Mozilla:
www.mozilla.org
Very netscape-like, but infinately more powerful. I hated Netscape too, but Mozilla 1.0 changed my mind (Tabbed Browsing KICKS!). Only problem is it can be a resource hog, especially in Win 98/ME (I use 98).
Firebird is the next-gen Mozilla that will replace Mozilla once finished. It's smaller and very fast, but the plug-in support is a bit weak right now.
Opera is a very nice browser and I use it sometimes. But the banner gets in the way, and I miss the tabbed browsing when I use it (even though it's not too much of an issue).
There are others too (Galeon, Netscape, Amaya, etc), but those are the ones I use the most. I rarely use IE anymore, although I do turn to it when I don't want to wait 6 seconds for the stupid thing to load (which is an advantage of IE over the other browsers, abiet a pretty unfair advantage but an advantage noneoftheless).
-edit- Mozilla does have problems rendering alot of pages right, but it's nothing you can't deal with. Firebird seems to do a better job, BTW.