OP, the main distinction (and what Old Hack said) is that there are two general groupings: the big five publishers -- and this includes their imprints (e.g. St Martin's is an imprint of Macmillan, a big five publisher) and everyone else. Those not among the big five and their imprints are considered "independent."
But then it gets tricky, the lines blurring. As Old Hack says, the independents vary in their size, plus they can go from independent to part of the big five in the blink of an eye (e..g, Harlequin, a large independent owned by Torstar, was sold recently to HarperCollins).
"Indie" is a term I now largely disregard. Old Hack uses it to denote "independent", but some people throw that term around differently, including presses that allow to people to self-publish. They basically equate "indie" with "alternative", which is not uncommon in the music industry but a bit of misnomer in publishing. I've seen on certain blogs where writers mention that they've been "indie" published and when I look at their books on Amazon, it's clear it's self-published. Some only available as e-books.
So, the first question is whether the press is independent of the big five or not. Then, as Old Hack says, how many titles do they put out annually give me a rough idea of whether they're mid-sized, small or micro.