|MadSci Network: Astronomy|
Your questions are very good ones and are a topic which is currently being debated among astronomers. Despite there being so many stars in the sky, the distances between stars are so great that it is very unlikely that two stars will wander by each other and collide. Instead, it is much more likely that the two stars will form together in the first place to create a binary star system. Usually, the two stars are separated enough so that they just orbit around each other for a really long time like the planets do around the Sun and not much else happens.
[The one important exception is for globular clusters and the center of the Milky Way Galaxy. Here the stars can be so close together that they can actually collide. Hence, stellar collisions are one of the popular explanations for blue stragglers in globular clusters. Blue stragglers are not seen in the Galactic center because there is so much dust between us and it, we cannot see most of the stars in the Galactic center.]
However, sometimes the two stars wind up orbiting so close that they do, in fact, interact with each other. We are not able to see much of what is happening day to day between two stars with our telescopes but luckily we do know a little about what might happen thanks to the laws of physics.
There are two important factors to consider when the two stars interact:
The masses (or gravities) of each star, and
How large the stars are.
The higher the gravity (mass) of a star is, the better it is at holding onto the gas in its atmosphere. However, massive stars are also very large (have bigger volumes), and the larger the star is the most likely it is to lose its atmosphere. This is because the material is so far away from the central massive core of the star that is not attached to the star by gravity as well as if the atmosphere were closer to the star.
With all this in mind, when two stars are interacting, material from one of the stars will actually be pulled off of it and go into orbit around the other star. This is called "mass transfer." The fact that the one star is loosing its atmosphere causes it to evolve much differently as if it were by itself.
Now, the other star has to put up with all this material which is now orbiting around it. How it reacts to this depends on the type of star it is. If it is a "normal" star like our Sun then it will just accept the matter from its neighbor and become more massive. This might be the reason for the existence of "blue stragglers" which are stars that are more massive than you would might expect given the ages and masses of their stellar neighbors.
If the star receiving the material is a white dwarf then we have what are called novae systems. Novae (not to be confused with supernovae) are bright flashes of light which occur when the material being sucked off the orbiting star hits the white dwarf and ignites. A white dwarf is the very dense core of a dead star which had a mass similar to our Sun.
If the star receiving the material is a neutron star or black hole, then we have an X-ray binary system. Similarly, the high energy X-rays are produced when material from the orbiting star falls onto the neutron star or into the black hole. Neutron stars and black holes are what can be created after a star much more massive than the Sun goes through a supernovae explosion at the end of its life.
Usually, we don't think about the two stars colliding into each other but that one stars gets "cannibalized" or eaten up by it more massive companion star. However, one really popular topic many astronomers are thinking about are the mysterious "gamma-ray bursts." These are really high energy bursts of light which we are detecting with many different types of telescopes and yet cannot identify the origin of the light. One explanation for gamma-ray bursts is that they are caused by two neutron stars or a neutron star and a black hole spiraling into each other. Right now we are relying more on mathematics and physics to help explain why we see the gamma rays because we are unable to take pictures of the stars while they are spiraling in toward each other. Cool, huh?
I hope this is helpful for you. The topic of stars colliding is exactly what many astronomers are wondering about themselves.
Try the links in the MadSci Library for more information on Astronomy.