First you need footage of an anime, either by downloading or obtaining it from DvDs (I can't explain how to get those due to site rules and the fact either way is illegal in most countries, especially the US), and a Video Editing Suite.
I personally used Windows Movie Maker to make my AMVs, but there are quite a number of different programs out there, and alot of them are not free (Adobe Premeire, Sony Vegas to name a view.)
There are tutorials online (use google) on how to make AMVs, but I will give you two words of advice:
1. DO NOT USE SUBTITLED FOOTAGE. This is concidered amaturish and will turn alot of people off your AMVs. As anime fans, its second nature for our eyes to snap to any subtitles on the screen, taking away from what we really want to see, Your AMV.
2. A slide show of pictures put to music does not an AMV make.