I’ve had reasonable luck with CamStudio and Xvid codec. Quality at 70% and auto-adjust framerate. It doesn’t tax the CPU too much to compress, so the computer usually doesn’t lag too bad while recording. (It can though if the other app you’re recording is processor-heavy.) Probably old news to some people, but it works.
Only odd bug with CamStudio is that longer videos tend to have audio desynchronization. (Just enough to be annoying, and I’m surprised it still hasn’t been fixed yet.) However that’s easily fixed by using the VirtualDub video editor. In VirtualDub it’s a matter of selecting all the frames, and setting framerate so video and audio matches, and saving out a second file.
Both CamStudio and VirtualDub programs are free and have a small footprint in terms of filesize or memory used while running. If I recall no install is needed to run either. Just have them on a memory stick and you should be good to go.
Not sure if it’s the “best”… But I’m used to it, and I’ve found it reliable enough. I also might have to try some of the other ones mentioned here, as it’s been a while since I looked for screen recording software after finding something I consider acceptable.