Aerobars are used by racing cyclists to get into a more aerodynamic position and thereby go faster more efficiently. They were first used by triathletes and some time there after adopted by road cyclists for use in TT's only (it is dangerous to use them around other people). Greg Lemond's controversial use of them in the TDF, beating Laurent Fingnon in the final TT by 58 seconds was the start of their common usage in road racing.
At some point recreational cyclists adopted them in order to give themselves more hand/back positions during long rides. You are most likely to find randonneurs and other long distance cyclists who do most of their riding solo using them. It is unwise to use them when riding with other people. They change your center of gravity on the bike and make the front end less stable - more prone to swerving. They also take your hands away from your brakes, therefore making stopping and slowing more difficult as well.
There are "clip on" aero bars, which attach to regular handlebars and integrated aerobars which are a dedicated one piece aero-bar - bull horn combos - sometimes even with the stem also built in.
They do change the way your bike handles, so it may take you some time to feel comfortable using them. It is inadvisable to try to take a sharp or fast corner while on aerobars. Go out to your hoods when descending or turning.