The term "independent game", or "indie game", generally refers to a video game developed without the backing of a major publisher.
The standard way that new video games get developed, currently, is that a publisher finances a development team to create the game, and the publisher also handles things like publishing the work, promoting/publicizing it, and any intellectual property issues that might come up.
However, some games are not produced this way. Some games are created by developers on their own time and/or cost, and typically published by the developers themselves (where "publish" sometimes just means putting up a web page with a description and a download link). Such games are developed independently of major publisher backing, and thus called "independent" or "indie" games. Indie games are almost always computer games (since software development kits for video game systems can cost quite a bit), are often simpler and/or of smaller scale than typical commercial games, and have less in the way of graphical and sound assets (as there's less money available in development to make highly detailed environments, textures, and such). Of course, this doesn't reflect on quality; the actual quality of indie games ranges from embarrassingly bad to embarrassing the big houses.
Nor does "indie" necessarily imply anything about who developed the game or how it was developed. Cave Story is a labor of love wholly created by a single amateur developer over the course of five years, and its success has been largely due to fans' word-of-mouth promoting, while Bastion was created by a team of several experienced developers who split off from the prominent video game company Electronic Arts and then published by Warner Bros. Interactive.
Many indie games have been released as freeware games, since the process of charging people money for the software can be a hassle and rather costly (due to transaction fees for credit cards or Paypal), and furthermore, it's easy to get people interested in your work if you offer it to them for free (and there's no shortage of people looking for free games on the internet), which is especially a good thing if you're an up-and-coming talent wanting to get recognition. Additionally, the recent rise of digital distribution services (such as Steam, most famously) has made it easier for indie developers to receive financial compensation for their work from their fans. Indie games are typically lower-priced than conventionally-developed games, and especially in comparison with so-called "AAA" ("triple-A") titles that are developed with big budgets and expected to be massive blockbuster hits. In theory, the lower budget/risk and lack of Executive Meddling for indie games also permits more creative risk taking, whereas failure or even mediocrity of a game can doom a studio tasked with making a multi-million dollar game (even in good economies, statistics of under-performing AAA studios being shuttered or undergoing large layoffs are common).note
In recent years, independent game development has been on the rise, in part due to the success of various notable indie games (most notably Cave Story and Minecraft), as well as the greater availability of digital distribution. This is aided by the rise first of laptops then of portable devices such as tablets and smartphones. The smaller games indie studios are capable of producing often run well on these lower powered devices where AAA studios are typically rereleasing their games from five or ten years ago for the same devices.
See also Doujin Soft - in Japan the term "indie" refers only to games sold commercially, while "doujin" is used to describe hobby projects that are available for free or sold only at conventions.
There are also Tabletop RPGs know as "indie games", but precisely what that means is up to debate. Almost any definition includes many games not commonly identified as "indie".