Not too long, download the FLV from YouTube, convert it (or a part of it) to GIF with Total Video Converter (not free) and, if needed, cut it more precisely in Adobe Photoshop (not free)... There might be free tools for all this stuff... I'm also aware of GIFSoup but I've never used it, I think it's free but it leaves a watermark on your GIF images.
*EDIT* Downloading the FLV from Youtube can be done easily with many tools. My favorite is Firebug's "Network" panel, which I use as part of my work (so why install something else), but there are a lot of Greasemonkey script which can do it too.
Total Video Converter. I tried like 3-4 other free utilities but they were all pretty horrible. Is there something better/easier I should be using?
For some reason I got one of these (even though my one computer has a tv card - really don't know why, probably because it was on sale for way less). Anyways, the software that came with it works pretty well for capture and edit. Depending on where it goes from there I can convert things with Total Video Converter
Yeah, like I Said; I've never done it that way myself.
I bought and use a program called Total Video Converter for this purpose myself, but assumed the OP was looking for something FREE as most people often too cheap to buy anything.