I use one called "Debut Video Capture Software" by NCH.
It can do full screen recording, partial screen recording, webcam recording, it can also record just the audio that is coming through the speakers (useful in a pinch for quickly scraping SFX).
The free version has a trial period, but once the trial period ends the only thing it does is force you to save all recordings to WMV format (this is what it does on the Windows version, at least). It has no recording time limits or forced watermark or anything like that. I actually converted an old VHS tape to digital using this.