Using the following definition, based on Wikipedia
Indie music originated in the United Kingdom and the United States in the 1980s. Indie rock is extremely diverse, the influences and styles artists has been extremely diverse, including punk. psychedilia, rock and country. Originally used to describe record labels, the term became associated with the music they produced and was initially used interchangeably with alternative rock.
Further from Wkipedia is Allmusic's description of indie;
Allmusic identifies indie rock as including a number of styles that are: "too sensitive and melancholy; too soft and delicate; too dreamy and hypnotic; too personal and intimately revealing in its lyrics; too low-fidelity and low-budget in its production; too angular in its melodies and riffs; too raw, skronky and abrasive, too oblique and fractured in its song structures; too influenced by experimental or otherwise unpopular musical styles.
For myself I would describe indie music is post rock with lyrics with a strong crossover with modern progressive rock, where you take the standard rock instruments, but do not use them to make the usual song structures associated with rock.
To start off I would recommend two obscure bands, from Northern Ireland Mojo Fury who I saw for the second time last night supporting ASIWYFA, the previous time was supporting Oceansize and also try Clownparlour the more experimental side of that band. Then from Philadelphia Ape School who are bravely experiemental whilst still making great catchy music.
My overall favourite indie bands would be Broken Social Scene, Elbow, Gomez, Kings Of Leon and Shearwater.
Indie has produced some of the most interesting and diverse music out there and so is worthy of promoton and appreciation.