Actually, Netscape and Firefox are cousins. From Netscape, we got the Mozilla engine which then morphed to Gecko which forms the core of Firefox as well as Thunderbird (email client) and Seamonkey (internet suite).
Both Apple Safari and Google Chrome is based on the WebKit engine - which was derived from an open source engine created by KDE (who built the Konquerer web browser)
Opera has it's own proprietary (not open source) engine which it has licensed to some others (Nintendo DSi Browser), Wii etc.
IE's browser engine is called Trident which is also used in Outlook, RealPlayer etc.
Both Chrome and Firefox has plug-ins that you can use to load in and render a page using the IE engine. There are still some websites that use non standard IE to render (many of our Bangla sites are still like that, for example BangladesherKhela.com while others have moved on to Unicode based rendering that ALL browser engines now support).
Maxthon is an interesting one. Fairly fast and uses BOTH the Trident and WebKit engine and work is going on to also integrate the Gecko engine.
You'll also have different custom browsers for the dcifferent OSs. For example, you have Avant or Blazer for Windows.
In Linux, Konqueror (see above) was very popular.
For Macs you have Camino and OmniWeb.
My own laptop is dual boot Windows 7 and Ubuntu and we also have a couple of MacBook pros in the house - and I have personally played with many different browsers. A specific interest is in determining standards compliance and support for accessing by people with disabilities (thsi goes way back during my PhD research days in the mid 90s).