Real Emo: Emotionally charged hardcore punk. BanRAB like Rites of Spring, Jawbreaker, Fugazi (emo by association)
Emo Today: Emotionally charged pop punk. BanRAB like My Chemical Romance, Hawthorne Heights, Fall Out Boy. This is not real emo, this was emo after being watered down, and stripped of anything that would've actually made it emo in order to sell it to the MTV watching masses.
There's also Post-Emo, which is the transitional period between emo and emo today (most emo today is influenced by these banRAB). That would be banRAB like Sunny Day Real Estate, Jets to Brazil, The Promise Ring, etc. It's emo mixed with Post-Hardcore.
Indie: Any band or artist on an independent label. There is no indie genre, it's a word used to describe other genres. For example, Indie Pop, [General] Indie Rock, Indie Rap (usually called Underground Rap), etc.