Nice question:
AAA or tripple A games are like the Hollywood movies, mainly big budget, big teams, big marketing, big everything, they are the monsters of the industry. Most common big companies produce AAA games. Prince of persia, Half Life, Assasin's Creed, WOW, you name it, are AAA games, with a huge team and a massive money capital. 400+ worked on Assasins Creed 2 and a bigger team is working on new titles.
Indie games, short for independent are games made usually in houses, or small studios, usually by individuals and not companies and usually the slang is : Garage companies.
The aim of an indie game is the feeling, the art, the message, the immersion, they want to take the player to a different level, make him feel smth different through a game. Thatgamecomany is an independant company that grew with time and 10 people are working there now, maybe you know thatgamecompany after the amazing psychedelic game they released: "FLOWER".
quoting the about us section:
"thatgamecompany designs and develops artistically crafted, broadly accessible video games that push the boundaries of interactive entertainment. We respect our players and want to contribute meaningful, enriching experiences that touch and inspire them.We seek talent that values integrity and personal growth within an environment of intense collaboration and experimentation. "
and Jonathan Blow the creator of Braid one of my favorite indie games
"Games are an important medium for communication. They have a mode of expression that's very different from what other media can do. However, the mainstream industry does not spend much effort exploring the expressive power of games; that's where the indies come in. Indies can make whatever games they want to make, because they feel those games are interesting -- which is something the mainstream industry is no longer capable of."
- Jonathan Blow, Number-None (Braid)
Take AAA games and big companies as a mass production agency working intuitively just to finish and go back home with cash in their pockets, and indie games as 1 artist working on a panting, it's his baby, it's his passion, it's his love and emotions.
i'm a dreamer, passionate about emotions, immersion,
Playing games such as Braid, Trine, Flower,Machinarium, World of Goo and many others made me see things, and feel, where the game has a message and not an image. Big companies are limited, limited by a story, a license, a contract with some bigshot movie or star. indie are free, flexible and full of imagination, nothing stops them from experimenting, going crazy, and innovating.
I guess that answered your question :D (and maybe explains why the AAA industry is not for me)