Open Source and Free Pixel Editors

Piskel
http://www.piskelapp.com/
A simple online web-based (HTML5) Sprite and Pixel art Editor. Offline version is also available.
Source Code: https://github.com/juliandescottes/piskel

piq
http://piq.codeus.net/
A free online (Flash) app for creating pixel art.

See also
http://pixelartor.com/
http://pixieengine.com/
http://www.pixel.tools/

GIMP
http://www.gimp.org/
The famous open source alternative to Photoshop.
See this note for using GIMP as a pixel art editor.

Paint.NET
http://www.getpaint.net/
A free powerful image and photo editing software.

GrafX2
http://pulkomandy.tk/projects/GrafX2
A bitmap paint program specialized in 256-color drawing.
Source Code (GNU GPL): https://code.google.com/p/grafx2/

mtPaint
http://mtpaint.sourceforge.net/
A painting program to create pixel art and manipulate digital photos.
Source Code (GNU GPL): https://github.com/wjaguar/mtPaint

Pixen
http://pixenapp.com/
A open source (but not 'free') pixel art editor for Mac OS X.
Source Code: https://github.com/Pixen/Pixen

Aseprite
http://www.aseprite.org/
An open source animated sprite editor & pixel art tool.
Source Code (GNU GPL): https://github.com/aseprite/aseprite/
Note: Aseprite is open source but not 'free' since donation is required for downloading the pre-built binaries. However, it is open source so you can compile for the binary by yourself.

Links:
http://pixelartus.com/tagged/pixel-art-tools