There are two different types of plates that make up the Earth's crust: continental crust, and oceanic crust. Oceanic crust is heavier than continental crust, so when the two collide head-on, the oceanic crust dives beneath the continental crust, in a process known as subduction. The oceanic crust melts as it enters the mantle. The magma formed, along with the way the continental crust is pushed upward, creates mountain ranges. This is occuring right now along the west coast of South America.
When oceanic and continental crust collide at angles, a different type of process can also occur. Transverse faults form where two plates are sliding along each other in opposite directions. One of the best examples of this is the San Andreas Fault in southern California. West of the fault, the state is on the Pacific plate, moving north, while east of the fault, it is on the North American plate, moving south. At some point, thousands or millions of years in the future, California will end up in Alaska. Right now, though, this transverse fault is a worry, as it is the reason California is so prone to earthquakes. But, more on that later.
Occasionally, continental crust collides with continental crust. In this case, one of the plates has to give, but niether wants to subduct. It ends up acting similar to a car collision, where the crust crumples as the two plates collide. A collision of this sort is occuring between India and Asia, and is creating the Himalaya mountains.
Finally, oceanic crust almost never collides with other oceanic crust. Instead, oceanic plates move away from each other. This creates a gap, through with magma from the mantle seeps up. The mid-oceanic ridges are actually the boundaries between oceanic plates!
A view of the plates on the modern Earth
There's a lot more to plate tectonics than I can fit into a post. So, for more information on how plate tectonics works, how it has affected the planet in the past, and how the Earth might look in the future, I recommend the Paleomap Project.