Indie: Dullness, tweeness, songs about spatulas and Neitzsche. Often some variation. Generally better than what's on the radio. See the Shins, the Flaming Lips, Iron and Wine etc.
Goth: Deep, haunting atmospheric music. Ranges in styles from techno to neo-classical. Some goth bands (Siouxsie and the Banshees for example) got pretty big in the eightries. Got mucked up with industrial and metal in the nineties. Some kick-ass bands at different ends of it include London After Midnight, E Nomine, XII.Stoleti, Inkubus Sukkubus, The Sisters of Mercy, Dead Can Dance, :wumpscut:, Bauhaus and Nosferatu.
Khar: Thinks goth is better than indie.