Welcome. Here's how this stuffarooni works.
Enter the Hall of Fame and unlock Kiloude City, then Friend Safari is available.
Post your IGN & friend code (3DS main menu > friend list). Ask someone what your mons are.
KEEP YOUR INFO IN YOUR NAME.
Once mutually added you may encounter the first two Pokemon in your friend's safari.
Both of you need to be signed onto the PSS for the third.

THE FRIEND LIST LIMIT IS 100, KEEP THIS IN MIND.
If you delete friends, they keep access to your safari, but you lose access to theirs.

Don't like your safarimons? You'd have to factory reset your 3DS to change them.
No worries, someone will need your safari!

Pokemon in Friend Safari have
>An increased shiny encounter rate
>Two perfect IVs AT LEAST
>Hidden Abilities (When all three mons are unlocked)

List of Pokemon in Friend Safari:
http://www.serebii.net/xy/friendsafari.shtml

Spreadsheet for your Safari tracking needs (UPDATED 9/9):
https://www.sendspace.com/file/uuuq3o
Obsolete Thread: >>21527511

TOTT: Will the original safari return in the remakes?