Windows comes with Movie Maker and Mac's come with iMovie. On Linux, there are a few programs that may be packaged with a distro. But, if you're going to get serious about video, then you need to kick it up a notch. Basically, go Hollywood, man!

Mac's

Final Cut rules here. There's the Pro version and the HD version. One is very expensive and the other is reasonable. In both cases, you get a lot more control over your final output. Recently, Adobe ported Premier Pro to the Mac platform. In many ways, FCP and Premier are similar (screen layout).

Windows

Lot's of choices here - Adobe's Premier Pro, Avid's offerings, Pinnacle, and Sony's Vegas. Prices are all over the place. One program that had been quite interesting in that it was inexpensive and very powerful came from Magix. It's a free trial download and has lots of powerful features for a low-cost program. Very understated.