In addition to many of the above you could check out Snapshot -- it's a neat little plat-former with a mechanic based around taking photos to move parts of the scenery.

It's not a platformer or side-scroller, but my wife and I also really liked The Binding of Isaac, and have spent hours playing through it and unlocking many of the ridiculous number of items, additional bad-guys and achievements.

I used to play a lot of flash games of that sort. I'm not completely certain these are all indie games, I apologize if some of them aren't.

Flash Games:

• The Company of Myself
• Shift 1, Shift 2, Shift 3
• Loved
• House of Dead Ninjas (Not a puzzle game at all. It has a sequel, Super House of Dead Ninjas, which is on steam)
• Depict 1
• Luftrauser (not a puzzle platformer, but whatever - I really like it!)
• Continuity
• Enough Plumbers

Non-flash Games:

• The original Spelunky (don't think it counts as a puzzle game though?)
• Antichamber (a 3D puzzle game)
• Hero Core (not a puzzle game either, 2D side scrolling shooter)

gah, I know there were dozens of fun indie platformers I played over time, but I don't have a list of them : ( I found most of them (and these) on the indiegames blog and TIG Source, (and by browsing kongregate). I'll try to remember to look when I get home to get a more complete list.

Not mentioned yet: Unepic

Dwarf Fortress

Brogue

Legend of Grimrock

Knights of pen and paper

Hotline Miami <- Probably my favorite indie title to date

Super meat boy

Fez

Monaco

The Binding of isaac

Mark of the Ninja

Retro City Rampage.