Firefox is the cross-platform open-source browser developed by the Mozilla Corporation. Firefox is currently the de-facto browser choice for the majority of Linux users, as well as many Windows and Mac OS X users. Firefox is secure, claims to be fast (personally I find Firefox significantly slower than its competitors) and offers its users a plethora of browser plugins. You can download the current version of Mozilla Firefox from their website.
Swiftfox is an optimized version of the cross-platform open-source browser, Firefox. It offers most of the options of Mozilla Firefox while attempting to address some of the speed issues of Firefox by providing builds for different processors / architectures. Swiftfox makes use of Firefox plugins and the Mozilla engine. If you think Mozilla Firefox feels a bit sluggish on your Linux install you might want to download and install a Swiftfox build tailored to your specific system.
Opera has been around since 1996 and has managed to survive the "browser wars" over the years and maintain a solid fan-base. It has evolved from a basic Internet browser to a full Internet application suite over the years. Opera is a solid alternative to Mozilla Firefox and constantly being improved. Get Opera here.
Google Chrome is a newcomer to the Linux browser scene. It is currently in beta but is already making a name for itself due to its speed and the potential of its plugin API (which is still a work-in-progress). For those who love Google Apps or are looking for a nice alternative to Mozilla Firefox, Google Chrome may be the browser for you. For more details on Google Chrome on Linux click here.
SeaMonkey is yet another Internet browser based on the popular browser code by Mozilla. The difference here is that SeaMonkey attempts to offer users a complete all-in-one Internet application suite. SeaMonkey includes an Internet browser, e-mail client, IRC chat as well as a newsgroup and RSS feed reader. For more information on SeaMonkey click here.
Epiphany is a very basic Internet browser for the Gnome desktop. Its focus is strictly on speed and simplicity. While it can handle basic web surfing tasks easily, it does not support all popular plugins (it does include a community maintained package of plugins). Epiphany is included with the basic install of the Gnome desktop.
Internet Explorer (via WINE)
Microsoft's Internet Explorer browser can be run via WINE or virtualization on Linux. However, unless you have a very specific reason for needing to run Internet Explorer on Linux, I recommend staying away from it. On Linux or on Windows, it's still nowhere near your best option for a web browser. Steer clear unless absolutely needed.