They drop in boxes from sales or you can buy them in the auction from players that are selling them.
Once you have one, rightclick the avatar ticket/item and it becomes active.
Avatars provide increased stats and extra EXP bonus. They are available in various levels, with increasing stats and EXP bonus. For example, a level 10 Avatar provides +21 stats and +35% EXP.
Use Alt-E to access the Avatar window. You will see 3 Avatar slots. You can have only one active Avatar. There is no practical reason to load an Avatar into one of the other 2 reserve slots. In fact, this can be dangerous— if you were to switch to an Avatar in one of the other slots, your prior active Avatar would be removed from the system!
Avatars of the same level are available in different stat combinations. For example, a Level 10 - P power Avatar provides 15 Power stats, 3 Skill stats, and 3 Impact stats. There are also Level 10 - S skill and Level 10 - I Impact types.
Avatar duration is measured in days. A typical avatar card is for 14 days and can be extended with Avatar Extension Tickets, although doing so costs more than the Avatar, unless it is a level 10 avatar you are extending.
The name of an Avatar is not important. For example, there is no difference between a Level 10-I Paula Creamer avatar and a Level 10-I K.J. Choi avatar.
You can choose, in the Alt-E window to turn of the Avatar appearance (e.g., the likeness of K.J.Choi). Turning off the appearance returns your character to its normal appearance and does not affect the stat and EXP bonus of the Avatar.