To my understanding an indie kid is someone who is not afraid to show their individuality. Generally, they dress in vintage or thrift store clothing, wear moccasins, or really anything that isn't quite main stream.
They also listen to mainly music that is classified under "indie rock" or "alternative". Typically signed by an independent music label.
Some bands that they commonly listen to are Death Cab For Cutie( a band popular amongst others that aren't indie), Of Montreal, Little Man Tate, Bromheads Jacket, Mumm-Ra, ect.
Though, that is not the only music they listen to, and usually they aren't completely against large labels despite the fact that many people like to make it seem this way.
Don't confuse a true indie kid with a poser or hipster (if that is what you prefer to refer to them as)
Hipsters like to advertise their individuality by going against anything completely main stream
Indie Kid: Wow! I really love Milburn, don't you?
Hipster: Yeah, they are so indie and different! Not like that crap that Katy Perry tries to sell.
Indie Kid: Uhmm... Okay, I kinda enjoy Katy Perry as well.
Hipster: Well, her music is the same as everyone else's! You're not a real indie kid!
Indie Kid: Alright then.