Our home galaxy, the Milky Way Galaxy, has about 400 billion stars, and is shaped like a fried egg, with a flattened disk and a bulge in the center. Between the stars, low density gas and dust is present in large concentrations known as interstellar clouds. Orbiting around the Milky Way are two smaller galaxies, the Small and Large Magellanic Clouds. The Magellanic Clouds will eventually merge with the Milky Way. This will not be the first time such a merger has happened --- in fact, right now another small galaxy, the Sagittarius Dwarf Galaxy, is in the process of merging with the Milky Way. In addition, some scientists have claimed to have found evidence for a second past merger of the Milky Way with another galaxy known as the Canis Major Dwarf Galaxy, but that is controversial. When the Magellanic Cloud-Milky Way merger will happen is uncertain, but on the order of 100 million years or so.
When a galaxy merger happens, stars will not collide together, because the spaces between the stars are immense compared to the sizes of stars (to visualize this, imagine that the Sun were scaled down to be the size of a basketball. On that scale, the closest star would be on the other side of the Earth). However, although the stars won't collide, the interstellar clouds will, since they are large, though low density. When such clouds get compressed enough, they gravitationally collapse and form into new stars. If a lot of new stars form all at once in a galaxy, this is called a `starburst'. When two massive galaxies with large amounts of interstellar gas clouds collide, this sometimes causes a starburst. For example, the merging pair of galaxies known as the Antennae galaxies are currently undergoing a starburst (see the Hubble Space Telescope picture of this ). In the Antennae, both galaxies that collided were about the same size. When this happens, it is called a `major merger'. During a major merger, so many clouds collide that many young stars are formed and a starburst happens. When a small galaxy falls into a much more massive one, like the merger of the Magellanic Clouds with the Milky Way, it is called a `minor merger'. Probably over the lifetime of the Milky Way, many such mergers have happened, not just the mergers with the Sagittarius Dwarf and Draco Minor Dwarfs. Since not as many interstellar clouds are involved, the bursts of star formation are generally not as powerful, and not as many new stars form. One might not even call such an event a 'starburst', as the term is usually used.
An individual starburst generally happens over a relatively short timescale, astronomically speaking--about a few million years. However, a galaxy can undergo many starbursts in its life due to multiple close encounters with one or more other galaxies. Furthermore, one close encounter between two galaxies can trigger the formation of stars in different parts of the galaxies at different times--for example, star formation can occur at different times in the central regions of the galaxies, in the outer disks, and in long streamers of gas pulled out from the galaxy by tidal forces from the second galaxy. The timescale between the start of a 'burst' in one part of a galaxy and the the start of a 'burst' in another part of a galaxy can be very long, more than 100 million years. No one has yet exactly modeled the future merger of the Magellanic Clouds with the Milky Way, so we don't know exactly where and when new stars will form due to the merger.