Friend Safari is a new location-esque mechanic introduced in XY. So far only X and Y have them.
Here's how it rolls:
When you befriend someone on your DS, the game receives some data (unknown at the moment) using which it creates a new location in the game. This is the Friend Safari you got from your friend. All Safaris look the same:
It's the contents of these Safaris that makes them special. Although all look like a rectangular patch of grass, every friend you have would have a different Pokemon in his safari.
For example, you have two friends Jason and Blake. Jason might have a Safari with Magmar, Pyroar and Ponyta, while Blake might have one with Gyarados, Frogadier and Krabby. They're almost never the same. Friend Safaris are best categorized in types (Fire Safari, Water Safari, etc)
There are also some other things that make it different from just about any other place:
A chance to find Pokemon with Hidden Abilities. Many Pokemon like Gale Wings Fletchinder and Protean Frogadier can only be caught here.
The number of Pokemon here can vary. If your friend has not beaten the Elite Four, or doesn't even own an XY copy, only two Pokemon can be found. However, if he has, then three can be found.
Higher chance of finding a Shiny Pokemon in Friend Safaris, i.e a 1/512 chance which is approximately 8 times the normal amount. It is unaffected by Shiny Charm though.
Pokemon found here are always at level 30, and have two perfect IVs.
Friend Safari itself is a location in Kiloude City. It stores many different Safaris (per friend) that you've accumulated in time. You can access these locations by talking to the attendant there.
That's the gist of it. Hope I helped!