Recs.
Updated
SpecsUpdate
Pros
Pro Free, open source and cross-platform
OBS is licensed under GPLv2 with code available on GitHub. It's available on Windows, OSX and Linux.
Pro Has over 20 plugins that enhance functionality
OBS has over 20 plugins to choose from, including one's that allow adding video files and any other media supported by VLC as a source, integrating Twich chat and switching scene depending on what window is in focus.
Pro Supports Intel's Quick Sync for faster and more power efficient video processing
Quick Sync is hardware video encoding and decoding technology that's comparable in speed, compression ratio and quality to x264. Since Quick Sync is a dedicated hardware core on the processor die, it allows for faster and more power efficient video processing.
Pro Several capture types supported
There are multiple capture types that can be used to stream. By clicking in "Sources Window", you can add the type that you need. Windows Capture_, for example, as the name suggests, captures the windows. _Monitor Capture is similar but captures the window regardless of where it was, this way you can for example capture the whole monitor or a sub-region of the screen. But the most important capture type by far is _Game capture_, which allows users to record full screen captures of games.
Cons
Con Captures overlays even when told not to
This software has had a bug for several years where it will capture the Steam overlay and other overlays. This is problematic because your private conversations can be viewed by anyone watching your current or past streams. Privacy issues are a major reason to avoid this software.
Recommendations
Comments
Out of Date Pros + Cons
Pro Has a tool that figures out best stream settings
OBS webpage has a Stream Settings Estimator that helps figure out optimal network and game configurations.
Edit: This has since been removed.