What's the good software to capture anything that is going on at my screen to a video file that I can later on upload to let's say YouTube? I'm using Mac OS X 10.6 (Snow Leopard).
I think Snapz Pro X is the best of the bunch for video capturing.
Scratch that! Snapz Pro X is expensive and slow and rather outdated at this point. I've started using Screeny and love it. Integrates with CloudApp for uploading if you like, has a slick easy to use UI, and most importantly doesn't cripple my machine when recording a screen cast1.
1: This could also be the 12GB of ram and quad-core processor, but I like to think it's just a better written app.
Open QuickTime / File / New screen recorder.
See this video
As others have said, Quicktime X can record both sound and video. Another free option is Copernicus. I used that successfully under Leopard. I've not tried it with 10.6 but see no reason why it wouldn't work.