The characteristics that make a game "indie" are contentious and this tag wiki makes no attempt to provide a definitive answer. However, most indie games are independent by virtue of their degree of authorial involvement - that is, how much of the work is done by the author of the game.
The concept of independent role-playing games was heavily championed by the Forge website (2001-2012). A long-used definition was "the game is owned by its author, or creator-owned" (ref), which later was specified to "creator-owned and self-published" (ref).
However, the use of the term outside the Forge community may follow other ideas and principles. Some indicators of "indieness" include:
Authorial ownership - An indie game is typically written by the same person or small team that owns the game rights. An individual creating a game by his-or-her self is the prototypical indie.
Authorial publication - A game where the author and the publisher are the same person or entity.
Authorial production - A game where the author was responsible for the production of the game: design, layout, etc..
As noted, this is a contentious issue and edge cases abound. For example:
- The Dresden Files - dresden-files is an indie game by most measures, yet the author / owners of the game didn't do all the work - and it's a licensed property to boot! But the game was built by a small team without the support of a larger business entity and was published by that same team of author / owners.
- GURPS - gurps may have been an indie game at one time, being published by neither WotC nor White Wolf, and SJ himself still owns the whole thing. But he's no longer a primary author of GURPS material. So GURPS probably fails the "indie" test, even though it is owned by the original author.
Admittedly, counter-arguments could be for either game. But this explanation should serve to give some initial guidance to the newcomer.