Mozilla was Netscape's internal name for their Netscape Navigator web browser, and the Mozilla/X.Y version numbers followed their browser versions.
For whatever reason, Microsoft decided to use the Mozilla/X.Y as well (possibly to show they were compatible with that version of NN) but they included they included the e.g. "compatible; MSIE 6.0;" bit in brackets to show the actual browser. Other browser vendors followed suit and they now all seem to show Mozilla/5.0 for the current version.
MSIE uses Microsoft's Trident rendering engine.
Firefox and other Firefox/Mozilla based products use the Gecko rendering engine, which is open source.
Opera uses its own Presto rendering engine.
Konqueror uses KHTML, an open source rendering engine.
Apple took KHTML and developed it futher, calling it Apple WebKit, and supposedly pass changes back to the KHTML developers. Webkit is used in Apple's Safari and Google's Chrome web browsers.