If you're a windows user, WMM is the most easily-accessible software.
However if you want better capabilities and can afford a program in the range of $60-100 (which ANYONE should, really) - Sony Vegas Movie Studio, Magix Movie Edit and Adobe Premiere Elements are all in that price range and are full-featured editors.
There's no real difference in "for AMVs specifically". AMVs tend to use more a bit more in the way of editing tools than most kinds of editing, but it really depends on your editing ideas and style - I would say for 90% of people, there's nothing they would need that Premiere Elements doesn't have. If you're interested in making the next big 3D effects-fest however, chances are you will need something a little more complex.
If you've not done any kind of editing and not sure if you even want to, start experimenting with WMM. Else start with one of the three I mentioned - they're well-suited to most editing scenarios.
Recently I’ve been haunting myself.
The shocking thing is how easy it is to disappear if you really want to.
I wonder for the millionth time how long it will be before anyone misses me.