Short answer, nothing much for most of us:

• Migratory birds might get confused since they rely on the direction of the magnetic field.
• Magnetic field strength reduces to about 10% during the transition and is much more complex during the transition, with multiple north and south magnetic poles.
• More auroras. Would get auroras at the equator during the reversal itself.
• Solar storms would have more effect but Earth would still have a magnetic field, just a bit weaker.
• Astronauts in low Earth orbit would need more protection from solar storms
• Ozone layer damaged leading to increased UV radiation so humans would need more sunblock to avoid skin cancer. It would stress the populations of phytoplankton in the oceans but there is no evidence yet to support the idea that this could have major wider reaching effects. This is a matter of on going research.
• No mass extinctions as a full reversal happens every two hundred thousand years approximately and we have nearly all the same species on Earth as we had hundreds of thousands of years ago.
• Our atmosphere still protects us, as it is equivalent to ten meters of water in mass. Only the upper atmosphere gets increased levels of particle fluxes. The cosmic radiation you detect at ground level is due to much more penetrating particles than the slower moving ones in solar storms.
• The solar storms have large scale electromagnetic effects, and we would have to harden long range transmission cables but we should do that anyway as present day solar storms can go right through the Earth’s magnetic field protection when they are strong enough.
• We get increased erosion of the upper atmosphere by solar storms, but magnetic field reversal is so rare that this has no long term effects - it can’t strip away the atmosphere even on geological timescales. Solar storms can do that on Mars because it had no magnetic field for billions of years

HOW OFTEN DO WE GET A MAGNETIC REVERSAL?

First, full magnetic reversals, where the field flips and stays flipped, are rare, roughly every 200,000 years but sometimes with much longer gaps between them. The last one was Brunhes–Matuyama reversal 781,000 years ago.

But sometimes the magnetic field reverses temporarily, and then reverts to its original state again. One geologically recent example, the Laschamp event 41,000 years ago. This happened surprisingly quickly, around a century for the polarity shift, unlike a full reversal that takes thousands of years for it to reverse.

It was a complete reversal, not just a change in position of the pole. While reversed, the field strength was only 5% of our normal magnetic field, but it had North and South interchanged. It lasted for 440 years. Of that time period, the two reversals took up 250 years.

An extremely brief reversal of the geomagnetic field, climate variability and a super volcano

It’s not much different though whether it just flips for a short time or flips for a long time.

In this diagram the yellow dots track the motion of the north "virtual geomagnetic pole"

For a couple of science news stories about this research: An extremely brief reversal of the geomagnetic field, climate variability and a super volcano , Ice age polarity reversal was global event: Extremely brief reversal of geomagnetic field, climate variability, and super volcano

It remained reversed for a total of 450 years and the two polarity reversals took 250 years of that. That's very rapid on geological timescales.

For the detailed scientific paper: Dynamics of the Laschamp geomagnetic excursion from Black Sea sediments. This diagram is discussed on page 65.

So, it does seem it is something that can happen. Not just in a few years. But over a couple of centuries.

There have been other magnetic field "excursions" as these are called. Gothenburg magnetic reversal 11,500 years ago and Mono Lake magnetic reversal of 23,000 years ago .

This is a simulation of a magnetic reversal on supercomputer from 2010, just to give an idea of how it works, it's not just the magnetic poles moving, like turning around, the magnetic field would get complex in the middle of the transition. It would get pretty hard to use a compass, I'd imagine, need to rely on up to date maps of the direction of the magnetic field in whatever part of the world you were sailing in.

This is what it's like in the middle of a reversal:

Geodynamo

We are nowhere near anything like that at present.

CURRENT SITUATION, NO SIGN OF A REVERSAL

The South dip pole lies at a latitude of 64.28 degrees South, outside Antarctica, in the open ocean, also outside the Antarctic circle.

While the North magnetic pole is far closer to the pole, almost directly at it right now:

As you see the N. magnetic pole is continuing to move closer to the geometric N. pole and the S. magnetic pole is continuing to move away from the geometric S. pole.

In these diagrams, the blue is the geomagnetic pole - treats the Earth as if it were a dipole magnet. So the geomagnetic poles are diametrically opposite each other. The red dots are the dip poles - the point on the surface where your compass needle would point directly downwards or upwards.

More about it here: Magnetic Poles

There's also evidence that the magnetic field is getting weaker. But it’s been much stronger than usual for a while and so far it is not particularly low, just declining towards rather ordinary values

What it will do next is anybody’s guess. If you extrapolate that graph, it reaches 0 so a reversal after 1500 years. But there is no reason to suppose that it’s doing that. Even if it gets very weak, often you get “excursions” where the field gets weak, but then just restores itself in the same direction as before.

So there is no reason to suppose it will reverse based on the magnetic field strength so far. The magnetic poles are continually moving anyway and at present they are close to the poles and the magnetic field strength is normal.

See Magnetic Reversals

But it could happen. And we can get an idea of the effects, from studies of the last time it happened.

EFFECTS OF THE REVERSAL LAST TIME IT HAPPENED

There were increased levels of radiation, with increased levels of Beryllium 10 and carbon 14. See https://www.sciencedaily.com/rel...

(note, in case of confusion: the paper doesn’t say that the reversal caused the supervolcano eruption, it’s just that their research allowed them to research both events as they were close together in time)

We remain protected by the atmosphere, which is roughly equivalent in radiation shielding to ten meters thickness of water. So we don't need to be concerned we'll all die, like astronauts caught in a solar storm outside the shelter of the Earth's magnetic field. That can't happen.

Human beings have managed fine through many previous reversals. Anatomically modern humans evolved around 200,000 years ago, archaic humans 500,000 years ago, and earlier hominids have been around for millions of years, see human evolution

The weaker magnetic field during a reversal wouldn’t make much difference for the faster particles in cosmic radiation as these fast particles go straight through the Earth’s magnetic field anyway. And some particles are even accelerated by the magneticfield. The Earth’s atmosphere protects us from this, again shielding equivalent to ten meters thickness of water.

Theoretically the increased radiation levels from the slower particles could increase cloud levels (because radiation is supposed to help with cloud formation, similarly to the way they produce trails in cloud chambers) which could cool the Earth. The authors of that paper couldn't find a clear correlation of weather with the cosmic ray flux during the Laschamp event however (just summarizing what they way in their paper).

More generally, there’s no proven link between magnetic reversals and extinctions.

“During a transition the magnetic field at the surface of the Earth decreases to about 10% of its current value. If the geomagnetic field is a shield against energetic particles of solar or cosmic origin then biospheric effects can be expected. We review the early speculations on the problem and discuss in more detail its current status. We conclude that no clear picture of a geomagnetic link, a causal relation between secular magnetic field variations and the evolution of life on our planet can be drawn.”

Magnetic Polarity Transitions and Biospheric Effects (2010)

In more detail: in the summary conclusion section on page 157 of the earlier paper:The Sun, geomagnetic polarity transitions, and possible biospheric effects: review and illustrating model (2009) they conclude that the main effect would be generation of a natural hole in the ozone layer and this would stress the populations of phytoplankton in the sea, but that so far none of the recent studies have yet been conclusive enough to decide if this has cataclysmic effects on the Earth’s ecosystem.

• “A major atmospheric effect of polarity transitions is most probably the generation of a natural ozone hole due to enhanced SPE activity. This ozone hole is associated with a strong increase of erythemal weighted surface UV-B flux. .
• “The increase of erythemal weighted surface UV-B flux represents a clear stress on aquatic ecosystems such as phytoplankton populations. Using a simplified model of enhanced UV-B stress on such a population indicates a complex, nonlinear response of the population.

… “We conclude that many further studies on details of the suggested process chain and actual analyses of geologic proxies are necessary before a possible connection following the processes discussed can be confirmed. All recent studies do not yet allow one to decide whether a polarity transition is a cataclysm to the Earth system or not. “

This is an earlier 1980 paper with the same conclusion: Relationship between biological extinctions and geomagnetic reversals

More citations in the wikipedia article here: Geomagnetic reversal

SOLAR STORMS

A really major solar storm will break through our magnetic field whatever, so that’s not particularly to do with magnetic pole reversal.

There's no risk to humans. But there is a risk to the electricity transmission network and to satellites mainly. Ordinary strong solar flares aren't really a problem, there is enough warning and the electricity companies and so on can take measures to protect themselves. Impacts of Strong Solar Flares. We get those every so often, every decade or so.

But then - there's the possibility of a really big solar flare. There was a big solar flare back in Solar storm of 1859. Known as the "Carrington event" after an English astronomer who was observing the sun, saw some huge sunspots, and spotted an intense white flash from the sunspot group. The auroras turned night to day, people could read the newspaper by the auroras. Gold minors in the Rocky Mountains woke up and ate breakfast at 1 a.m. thinking it was sunrise on a cloudy day.

Telegraphs stopped working - and in the USA, some operators disconnected the batteries and found they could send telegrams just using the induced electricity from the storm. See Severe Space Weather Events Telegraph operators also saw sparks leaping from their equipment, some big enough to cause fires. What If the Biggest Solar Storm on Record Happened Today?
 
 So - at the time that was just a curiosity and hardly made any difference to anyone except the telegraph operators and people woken up early by the bright auroras. But if we had a storm like that now, the effects could be huge. We have never had a flare anything like that big since then.

The main effects are:

• It could knock out many of our satellites including GPS by interfering with its electronics. I think the main risk here is that they go off line and have to reboot - not physical damage Impacts of Strong Solar Flares.
• GPS becomes less accurate during a solar storm - you get warnings about that however beforehand Space Weather Effects on GPS and WAAS
• It could destroy the transformers in our electricity grid for transmission of electricity.

Basically the power companies need to install monster surge protectors. Solar Storms: What You Need to Ask Your Power Company

And another approach involving adding extra resistors - this amounted to a total cost of the order of $100, million, for an event that could cost trillions (between 0.6 and 2.4 trillion dollars to replace damaged transformers after sch an event according to the Lloyds report) and mean outages of electricity for between 6 days and years. An Inexpensive Fix to "Prevent Armageddon" But Congress didn't pass the bill that was proposed to spend this $100 million on this fix.

I'm not sure of the latest on this. There's a lot about this online but it can be a bit hard to sift the accurate sites from the ones that are a bit over the top and sensationalist.

Blackouts certainly can happen, this is something that actually did happen in Quebec in 1989 You are most vulnerable in the higher latitudes so the North of the US would be the ones who lose power, and the higher latitude countries in Europe. Apparently also more vulnerable if the power lines run above igneous rocks.

"Power systems in areas of igneous rock (gray) are the most vulnerable to the effects of intense geomagnetic activity because the high resistance of the igneous rock encourages geomagnetically induced currents (GICs) to flow in the power transmission lines situated above the rock. "
The Day the Sun Brought Darkness

And - is something you can do something about - ways of protecting the transformers in power grids seem the most important thing to focus on. There's a useful recent discussion here at physicsstackexchange:
Can a Coronal Mass Ejection (CME) cause a blackout on Earth and why?
Where one of the answers says that the power network has various unintended protections built in, mainly that if one transformer blows out, the rest in the grid tend to trip rather than blow out too. And that in a study that he and some colleagues did, they found the power grid may be less vulnerable than previously predicted because of these reasons, but satellites that orbit at geostationary orbit, also the middle level orbit GPS satellites may be more vulnerable than previously expected, with many of them, if on the sun side of the Earth (between it's magnetic field and the sun) likely to be destroyed.

"So the most recent idea is that our satellites are very vulnerable but our power grids may not be as vulnerable as we originally thought (though, all of these issues are incredibly difficult to model and predict so take my comments with a grain of salt)."
- see the conversation here: Can a Coronal Mass Ejection (CME) cause a blackout on Earth and why?

Any other links on this?

(This is a shortened version of Robert Walker's answer to How often do solar storms occur? Can they hit earth or cause harm to use?)

NO DIFFERENCE IN NUMBERS OF PARTICLES THAT GET DOWN TO GROUND LEVEL

Solar storm particles are too weak to get through the atmosphere at all. Cosmic "rays" actually particles (the name is confusing as they aren't photons or radiation and travel at less than light speed) - they can, but the atmosphere is equivalent to 10 meters thickness of water so only the most energetic can get all the way through.

The loss of magnetic field won't make any difference there as it’s our atmosphere that protects us most (though it would make a big difference to astronauts in the ISS). It increases the number of particles that hit the upper atmosphere, whch is why it can influence the ozone layer and perhaps cloud formation. It also makes magnetic field differences to the surface which is how you can get the effects on long cables such as electricity transmission cables. But it doesn't increase the number of particles that get down to ground level in the atmosphere.

AURORAS

You'd see auroras right down to the equator.

Here is a stunning video of the Aurora Borealis from the ISS in 2012.

And a compilation of various videos of it here

See also If you were alive during the reversal of Earth's magnetic poles, would you notice anything on Earth while it occurred, like a sudden change in weather?

and Aurora Borealis: Why is Antarctic Auroral Oval always off center over the South Pole?

This is identical to my answer to What will happen when the magnetic poles shift?