It's pretty easy, actually. An indie label is a business that operates along similar lines to a major, just on a smaller scale. But they tend to be incorporated, have a contract with a distributor, produce physical albums, etc. The first labels to put out early rock were indies. Same with the early LA punk scene, which included Black Randy's label Dangerhouse or SST. In San Fran, you had Jello Biafra's label, Alternative Tentacles. The British scene had several labels, including 4AD and the fantastic Mute. ETC. Go read The Big Takeover, and it's full of ads for still functioning Indie labels.
The big changes lately seems to be a shift towards labels that are centered on a single artist, rather than as acting as a place for a variety of artists, they are basically incorporation for one artist to put his or her work out. Amanda Palmer is an example of this. Young God's Michael Gira's label is more like this (at least I don't think he puts out aritsts other than his own work any more).
But plenty of artists just put their stuff online without necessarily becoming a business.They likely aren't trying to make a living off making music, but are just interested in sharing what they do with others, and if they make some money off it, that's great, but it's not their primary goal. I do think that it's messier now than it was in the 80s say, because now it's cheaper to make your own recordings and share them. But there is a difference.