The collision of another star with our sun would be a very effective way of ending all life on Earth. Fortunately stars are so far apart that the the chance of it happening in the life of the sun is very close to zero.
However in the densely packed cores of globular clusters the rate of collision is much higher, and there may be a collision somewhere in the milkyway once every 10000 years or so. Three way collisions don't happen (or are exceptionally unlikely) but would be similar to two two-star collisions.
The result varies depending on the size of the stars, their density, the relative velocity, and whether a collision is head on or not. Generally a denser star is likely to come out of a collision in better shape. So a red giant, could have most of its very rarified outer layers pulled off and the core escaping.
On the other hand, if two main sequence stars, like our sun, collide, they first compress each other. Some of the stellar envelope escapes perpendicular to the collision (think of jam being squeezed out of a sandwich). But within a few hours the two stars have mixed and merged, and a new, larger star formed.
The mixing adds lots more hydrogen to the core of the star, which causes it to rejuventate. The first evidence of stellar collisions is in globular clusters, in which there are some large blue stars that seem younger than all the other stars in the cluster (Blue stragglers)
A white dwarf and main sequence star collision is bad news for the star. The collision causes the core of the star to heat up and a massive release of thermonuclear energy. Plenty more than is needed to completely disrupt the star. The star is completely destroyed in the process. However the white dwarf is relatively unscathed.
White dwarf-white dwarf collisions can, depending on the size result in gravitational collapse to a neutron star.
The paper Stellar collisions, mergers and their consequences is a very readable summary.
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