I recently put up wire shelving in my pantry, and spent forever figuring out what type of drywall anchor I should use. I even found several types that I didn't even know existed. It might be good to have a reference page for drywall anchors. So, what type of anchors have you used, and for what projects?
I almost always use the self-drilling stud solver type anchors. I find that they are tough enough to hang just about anything.
Aside from those, the only other anchors I've found that I like are the toggle-bolt type, which are more useful if you need to hang something from the ceiling.
There are plastic anchors.
My suggestion, avoid them like the plague.
I much prefer the self-drilling that Eric posted, easier to install and much stronger hold in my experience.
With these, you have to pre-drill the hole, get it too small and it doesn't fit and you have to drill again, or it gets stuck partially and you have to rip it out or try to mash it in. Get it too big and the anchor doesn't do its job.
To get them to hold well, you have to get the perfect size hole.
Here is a similar answer from another question though this question is more multi-purpose.
There are several kinds of drywall anchors and they each have their own weight rating. Some work by drilling a small hole and tapping in a plastic sleeve and others work by drilling a bigger hole and screwing a plastic sleeve and there are others where you drill a hole and the metal butterfly expands behind the drywall know as molly bolts. Then there are toggle bolts that are spring loaded metal wings that fold and have a long machine screw. Recently I saw anchors where you drill a 1 inch hole in the drywall and this big contraption grips the inside of the drywall and is effectively a bigger version of toggle bolts (made by Moen and they're called SecureMount).
The biggest difference between each is how much weight they are rated to hold and how bit a hole you need to put into the drywall (and usually the bigger the hole in the drywall the more weight they can hold.
I also found this useful link that has pictures and describes all the types mentioned (except the SecureMount of course . . . those are new and very much a niche product).