These devices are are able to produce a wide array of particles and spatial effects to get a vessel out of tight situations. They were originally designed to "deflect" interstellar particulates (dust, micrometeoroids and the like) away from the ship while it is traveling at warp, but through the course of Star Trek: The Next Generation, Deep Space Nine, and Voyager, the system evolved into a "catch all" device, able to produce practically everything from resonance bursts to tachyon beams. They are the proverbial "Swiss Army Knives" of Federation starships.
In Star Trek: First Contact, a deflector dish was used to open a portal in time from the late 21st century to the 24th. The Borg also attempted to use this deflector dish to create an "interplexing beacon;" a device which they could use to contact their homeworlds in the Delta Quadrant.
In the Star Trek: The Next Generation episode The Best of Both Worlds, the deflector dish was converted into a weapon to be used against the Borg.