Where two tectonic plates come together is called a convergence zone, and many interesting and often violent natural displays are the result. Earthquakes, volcanoes, massive trenches and island-building all occur in the zone where plates collide. Often one plate slides beneath another, called subduction, and is recycled back into the Earth's interior. Alternately the plates simply grind against each other until something gives.
When plates come together it is known as convergence. There are three distinct types of convergence, the first of which is when an ocean plate pushes into a continental plate. When this happens, the former is subducted beneath the continent, causing a trench to form at the edge of the ocean plate and a mountain range to be created at the edge of the continental plate. Pieces of the subducting plate break off and become dislodged at various points along the subduction zone. When they eventually break loose, earthquakes are the result.
When an ocean plate runs into another ocean plate, a trench forms where one dives beneath the other. An example of this is the Marianas Trench in the Pacific Ocean where the Phillipine Plate is in the process of subducting beneath the Pacific Plate. Another characteristic of this action is the formation of undersea volcanoes on the ocean floor which, as lava and other volcanic debris piles up, eventually creates island chains, like Hawaii.
When continental plates collide there is usually no subduction involved. The relatively light material instead causes the event to resemble two colliding icebergs, where neither will give an inch. In this type of convergence, both plates will buckle, creating mountain ranges and setting the stage for earthquakes in the region. The Himalayan Mountains were and are still being created through this very mechanism, in this case a collision between India and Asia. In the absence of a subduction, continental convergence usually creates a sideways sliding movement as the plates continue to push against one another.