Flashpoints are instanced group content areas of Swtor. What that means, if you're new to MMORPGs is that upon putting together a group and entering the flashpoint you will be "instanced" off from the rest of the world. The only other players you will be able to physically interact with are your groupmates. There are currently only 15 available flashpoints, some of which are restricted to a specific faction. The flashpoints in the game cover a wide range of levels, the earliest one starting at level 10 and the hardest one being of course a level 50 flashpoint.
Flashpoints offer better loot than regular quests and a much greater challenge. Most of the flashpoints currently in the game are located right off the main level of your faction's fleet station, however there are of course some flashpoints in different locations. Swtor does good in reminding players that there are flashpoints available and waiting to be done by placing <Flashpoint Courier>s in almost every major town/planet throughout the game. To see what flashpoints are in your level range just visit one of these NPCs and speak with them. They are most commonly located in the Space Ports of each planet.
Want to make sure your character is all he can be before running Flashpoints? Be sure to grab all of the Datacrons in the game then!
(Republic) Taral V
(Imperial) The Black Talon
The Colicoid War Game
(Republic) The Esseles