During the collision as others said, almost no chance of stars colliding. To put it another way, stars are moving in many different orbits inside our galaxy already, and they don't collide usually or even come particularly close to each other.
This is an example of a fast moving nearby star.
WISE: New Stars and Brown Dwarfs
The stars in our neighbourhood and throughout the galaxy are moving in many different directions anyway - except ones that have a shared birth in a star nebula, which follow similar paths for quite a while. Stars are continually bobbing up and down above and below the galactic plane (attracted by the gravity of the disk) as they orbit around the galaxy on even longer timescales - so while our sun is going down, another one is going up and vice versa. Other stars are remnants of dwarf galaxies that were swallowed by the milky way in the past. And others are just old stars that have had many (very distant) flybys of other stars and so picked up a high speed from the gravity boosts.
During the collision then there will be stars that pass us by at higher velocities than usual but no reason for them to pass significantly closer than they do already. Some would surely get close enough to send more comets towards the Earth as has happened many times in the past (most end up hitting Jupiter or the sun), pretty much business as usual.
However, our sun will surely be sent into a new orbit around the core. It could even be sent into interstellar space and become one of the stars that wonder through the universe without a parent galaxy (and so safe from collisions from then on).
And our spiral galaxy will become an elliptical galaxy.
And another simulation of part of the process
When the sun starts to go red giant 5 billion or so years from now, then the Earth may survive, or may be swallowed up by the expanding sun. Either way though, as the sun gets hotter, it becomes uninhabitable by us some time before a billion years from now.
So our descendants or distant cousins by then will have to do something about it. Which could be to leave the Earth or could be to move the Earth - it's not beyond the possibility of technology to move it outwards to a more habitable position given billions of years to move it (e.g. by gravity tractor of repeated flyby of a large asteroid or perhaps by other methods also). Even to make the Earth into a spaceship and head off to another star system (perhaps using fusion power to create an artificial sun for the journey). With abundant power, e.g. from easy to use fusion power or even complete matter conversion if we achieve that somehow, or using self replicating machines, then things like that might be within our capability - or our cousins or descendants that is, a million times removed.
Later on - the end of the red giant phase, it throws off it's hydrogen as a result of the helium burn.
Like this cat's eye nebula (around a star that was about five times the mass of our sun originally). The nebula itself is mainly hydrogen or helium but it would be quite a violent climate on the sun leading to it throwing off material like this.
Cat's Eye Nebula
See Red giant - Characteristics - for details of what happens during the Red Giant phase
So if we survive that long, even if Earth has been moved out away from the sun, I imagine we'd probably be living in the oceans or somewhere well protected from all the solar activity of a sun in its red giant phase. Or could be, living in subsurface oceans of icy moons. Saturn's moon Titan could become an ocean world when the Red Giant is at its maximum. Or if we are able to move the Earth we could make it a Moon of first Jupiter and then Saturn - or else as a Trojan for them, and when at in-between orbits, just orbit the sun independently. Where resonances from Jupiter would make some of the orbits unstable - the mechanism that clears regions of the asteroid belt - but with this level of megatechnology to move the Earth those would not be a problem for us.
Finally our sun collapses to a white dwarf and those are stable for quadrillions of years just gradually cooling down. So then we could orbit close to it if still there and able to move the Earth again - only thing is - tidal effects would surely put us into an orbit always with one side towards the star. But that's survivable and if the planet has an ocean or atmosphere or both this can help maintain thermal equilibrium between the night and the day side.
An Earth orbiting a white dwarf can remain in the habitable zone for up to 10 billion years ir si depending on the distance and on the size of the white dwarf. See Transit surveys for earths in the habitable zones of white dwarfs.
It would have a "year" of around one Earth "day" - ideally you'd start at well over 40 hours, and then towards the end speed up the Earth's orbit to 5 hours to remain in the habitable zone. Eventually after 10 billion years, then the habitable zone gets too close to the white dwarf, within the Roche limit.
So if you are on the side facing away from the sun you will see the stars rise and set at about the same rate they do on the Earth. And on the "day" side their sun will be approximately the same colour and same angular size as our sun, because white dwarfs are similar in surface temperature to our sun (at 5000K).
A black dwarf is thought to take at least a quadrillion years to reach thermal equilibrium, and possibly much longer if it is kept warm by processes such as proton decay, see Black dwarf. It is a very slight warming, only 0.06 K, but by then the three degree background radiation has cooled down to even less than that. Page on lanl.gov
There's a good wikipedia article on this, summarizes some of the main events in our far future: Timeline of the far future
But anyway - not something we need to worry about at present. This is for some future creature to deal with, which may or may not resemble us, it might have completely different metabolism and needs and wishes and aims from us.
It's even possible that some creature that at present can only be seen in microscopes, such as this one:
Or if it is us, we may have changed radically by then also just through the process of evolution, if nothing else.
For more on this, see: Robert Walker's answer to What are the odds that a viable population of humankind will escape to another planet before the earth becomes uninhabitable?
This is an example of a fast moving nearby star.
WISE: New Stars and Brown Dwarfs
The stars in our neighbourhood and throughout the galaxy are moving in many different directions anyway - except ones that have a shared birth in a star nebula, which follow similar paths for quite a while. Stars are continually bobbing up and down above and below the galactic plane (attracted by the gravity of the disk) as they orbit around the galaxy on even longer timescales - so while our sun is going down, another one is going up and vice versa. Other stars are remnants of dwarf galaxies that were swallowed by the milky way in the past. And others are just old stars that have had many (very distant) flybys of other stars and so picked up a high speed from the gravity boosts.
During the collision then there will be stars that pass us by at higher velocities than usual but no reason for them to pass significantly closer than they do already. Some would surely get close enough to send more comets towards the Earth as has happened many times in the past (most end up hitting Jupiter or the sun), pretty much business as usual.
However, our sun will surely be sent into a new orbit around the core. It could even be sent into interstellar space and become one of the stars that wonder through the universe without a parent galaxy (and so safe from collisions from then on).
And our spiral galaxy will become an elliptical galaxy.
And another simulation of part of the process
When the sun starts to go red giant 5 billion or so years from now, then the Earth may survive, or may be swallowed up by the expanding sun. Either way though, as the sun gets hotter, it becomes uninhabitable by us some time before a billion years from now.
So our descendants or distant cousins by then will have to do something about it. Which could be to leave the Earth or could be to move the Earth - it's not beyond the possibility of technology to move it outwards to a more habitable position given billions of years to move it (e.g. by gravity tractor of repeated flyby of a large asteroid or perhaps by other methods also). Even to make the Earth into a spaceship and head off to another star system (perhaps using fusion power to create an artificial sun for the journey). With abundant power, e.g. from easy to use fusion power or even complete matter conversion if we achieve that somehow, or using self replicating machines, then things like that might be within our capability - or our cousins or descendants that is, a million times removed.
Later on - the end of the red giant phase, it throws off it's hydrogen as a result of the helium burn.
Like this cat's eye nebula (around a star that was about five times the mass of our sun originally). The nebula itself is mainly hydrogen or helium but it would be quite a violent climate on the sun leading to it throwing off material like this.
See Red giant - Characteristics - for details of what happens during the Red Giant phase
So if we survive that long, even if Earth has been moved out away from the sun, I imagine we'd probably be living in the oceans or somewhere well protected from all the solar activity of a sun in its red giant phase. Or could be, living in subsurface oceans of icy moons. Saturn's moon Titan could become an ocean world when the Red Giant is at its maximum. Or if we are able to move the Earth we could make it a Moon of first Jupiter and then Saturn - or else as a Trojan for them, and when at in-between orbits, just orbit the sun independently. Where resonances from Jupiter would make some of the orbits unstable - the mechanism that clears regions of the asteroid belt - but with this level of megatechnology to move the Earth those would not be a problem for us.
Finally our sun collapses to a white dwarf and those are stable for quadrillions of years just gradually cooling down. So then we could orbit close to it if still there and able to move the Earth again - only thing is - tidal effects would surely put us into an orbit always with one side towards the star. But that's survivable and if the planet has an ocean or atmosphere or both this can help maintain thermal equilibrium between the night and the day side.
An Earth orbiting a white dwarf can remain in the habitable zone for up to 10 billion years ir si depending on the distance and on the size of the white dwarf. See Transit surveys for earths in the habitable zones of white dwarfs.
It would have a "year" of around one Earth "day" - ideally you'd start at well over 40 hours, and then towards the end speed up the Earth's orbit to 5 hours to remain in the habitable zone. Eventually after 10 billion years, then the habitable zone gets too close to the white dwarf, within the Roche limit.
So if you are on the side facing away from the sun you will see the stars rise and set at about the same rate they do on the Earth. And on the "day" side their sun will be approximately the same colour and same angular size as our sun, because white dwarfs are similar in surface temperature to our sun (at 5000K).
A black dwarf is thought to take at least a quadrillion years to reach thermal equilibrium, and possibly much longer if it is kept warm by processes such as proton decay, see Black dwarf. It is a very slight warming, only 0.06 K, but by then the three degree background radiation has cooled down to even less than that. Page on lanl.gov
There's a good wikipedia article on this, summarizes some of the main events in our far future: Timeline of the far future
But anyway - not something we need to worry about at present. This is for some future creature to deal with, which may or may not resemble us, it might have completely different metabolism and needs and wishes and aims from us.
It's even possible that some creature that at present can only be seen in microscopes, such as this one:
Sphaeroeca, a colony of choanoflagellatesapprox. 230 individuals), in light microscopy. File:Sphaeroeca-colony.jpg - Wikimedia Commons
almost any kind of present day living creature, even something as microscopic as this, may be the ancestor of the creatures with technology of that far future that will face this problem.
Or if it is us, we may have changed radically by then also just through the process of evolution, if nothing else.
For more on this, see: Robert Walker's answer to What are the odds that a viable population of humankind will escape to another planet before the earth becomes uninhabitable?
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
4.8m answer views110.4k this month
Top Writer2017, 2016, and 2015
Published WriterHuffPost, Slate, and 4 moreHuffPost, Slate, Forbes, Newsweek, Business Insider, and Science 2.0