Okay, first I assume we are talking about a massive “supervolcano” eruption here, which happens rarely.

DISPELLING SOME MYTHS FIRST

First, it’s not at all certain the next eruption would be a supervolcano - indeed it’s far more likely to be an ordinary eruption. There have been 80 non explosive eruptions in the last 640,000 years since the last supervolcano eruption. So that’s the most likely eruption by far. The last 20 of those were mainly lava flows. An eruption like that would disrupt activities in the Yellowstone national park itself, but likely to lead to few deaths and would not be catastrophic.

The average of the two intervals between the last three major past eruptions is 740,000 years and that’s the basis of the often quoted 1 in 740,000 chance of an eruption per year - or 1 in 7,400 per century. But that’s not a very compelling argument. Some scientists think that it may not erupt as a supervolcano again ever, that it may be winding down in its activity.

Also our understanding of these large volcanoes has moved forward and with modern understanding they think that if it did happen, the build up to a supervolcano in Yellowstone would be detected weeks in advance, perhaps months or years. Volcano Hazards Program YVO Yellowstone

WHAT IF IT DOES HAPPEN?

However what if there is a supervolcano eruption, what happens then?

“The term “supervolcano” implies an eruption of magnitude 8 on the Volcano Explosivity Index, indicating an eruption of more than 1,000 cubic kilometers (250 cubic miles) of magma. Yellowstone has had at least three such eruptions: The three eruptions, 2.1 million years ago, 1.2 million years ago and 640,000 years ago, were about 6,000, 700 and 2,500 times larger than the May 18, 1980 eruption of Mt. St. Helens in Washington State.” Yellowstone Volcano & Supervolcano

Grand Prismatic hot spring, Yellowstone National Park. Estimate of 1 in 700,000 chance of an eruption per year (1 in 7,000 per century). which could kill 90,000 people. See What would happen if Yellowstone’s supervolcano erupted?

But how much effect would it have? Well the largest supervolcano was 2.1 million years ago. If it was as large as that, it would be very extensive in its effects. If it was as large as the one 640,000 years ago it would also have extensive effects.

It’s hard to know from studying the geology as most of the ash would be eroded and not preserved. But a recent study in 2014 used a computer model to try to figure out what the effects would be, assuming 330 cubic kilometers of volcanic ash, dense-rock equivalent (so much more thickness for the ash itself).

It’s summarized here: Modeling the Ash Distribution of a Yellowstone Supereruption (2014) and the paper itself is Modeling ash fall distribution from a Yellowstone supereruption

This graphic from their paper shows how the volcanic ash would spread out over the US after such an eruption:

You’d get 1–3 millimeters thickness of ash right out to New York, which is enough to “reduce traction on roads and runways, short out electrical transformers and cause respiratory problems”. There would be centimeters of thickness over much of the mid west, enough to disrupt crops and livestock, especially if it happened at critical time in the growing season. and a meter of thickness out to quite a distance. The worst affected in their list of cites is Billings, population 109,000, which their model predicted would get an estimated 1.03 to 1.8 meters thickness of ash.

Artist’s impression here:

SUPERVOLCANOES WORLD WIDE

First, how likely are they? There’s an estimate here of the probability of a supervolcano happening anywhere in the world, and its effects. They say that there have been 42 supervolcanoes in the last 36 million years. However those came in two pulses and the rate varies between 22 events per million years and 1.4 events per million years.

The worst supervolcano in recent times was the one that created Lake Toba in Indonesia about 75,000 years ago. It’s 100 kilometres by 30 kilometres, maximum depth 505 metres.

Danau Toba

Its ash covered Malaysia to a depth of 9 meters, there’s an ash layer from it in central India that’s still 6 meters thick today, and ash from it is detected as far away as Lake Malawi in East Africa.

It injected 2500–3000 km³ of debris into the atmosphere, and probably killed 60% of the human population worldwide, mainly through climate change impacting on their food supply.

That’s the picture generally, that the main effect is through global climate change, which reduces the temperature globally by about ten degrees C for a decade, together with the direct effects of the deposits of ash on their crops. A large supervolcano like Toba would deposit one or two meters thickness of ash over an area of several million square kilometers.(1000 cubic kilometers is equivalent to a one meter thickness of ash spread over a million square kilometers). If that happened in some densely populated agricultural area, such as India, it could destroy one or two seasons of crops for two billion people.

It would also mean that you can’t fly jets in the affected area for as long as the air is filled with ash, but that’s a minor effect compared with the rest of the devastation. As for the “noxious gases” such as sulfur dioxide - these mainly make a difference to the upper layers of our atmosphere by combining with water vapour to create clouds that block out the sun. It’s not like you’d have trouble breathing or anything like that globally - the amounts of gas are far too small for that, it’s manly the effect of dust and clouds on the climate.

We could prepare for this. With just a couple of years of warning, we could do large scale tests to work out which crops we’d need to grow in the cooler world for that decade, and store crops that are currently used to feed cattle or for production of ethanol, which would be enough to give us a buffer for the first year. In that way we would be able to avoid perhaps all deaths from starvation, so impact would be far less than on early man.

For details, see Extreme Geohazards: Reducing the Disaster Risk and Increasing Resilience from the European Space Foundation, and for the Yellowstone park eruption simulation,: Modeling the Ash Distribution of a Yellowstone Supereruption (2014) which summarizes Modeling ash fall distribution from a Yellowstone supereruption

This answer is based on those two sources, together with the USGS FAQ here Volcano Hazards Program YVO Yellowstone plus I used some details from the Wikipedia article on Lake Toba - you can check the citations there for more details.

I’ve copied this answer (with minor changes) to What really happens if Yellowstone erupts as a supervolcano, or if some other supervolcano erupts? by Robert Walker on Debunking Doomsday