UPDATED 11/01/10 --- While most web browsers are free and you could download all of them, trying each one, the download and installation time would be quite lengthy, and some of them probably wouldn't even work on your computer. Read on to find out which browser is most likely best for you...
GRAPHICAL WEB BROWSERS
This is the most common type of web browser which most people are familiar with, and allows viewing of the greatest number of webpages and images. If your computer has a 166 MHz or faster processor and Windows, Linux, or Mac OS, a graphical web browser is probably the best choice.
Internet Explorer 8.0: Microsoft IE 8.0 is substantially different than previous versions like 6.0 and 5.5. Like 7.0, it offers tabbed browsing (which was already popular in other browsers like Opera), provides a larger webpage viewing area, offers a convenient built-in search function, and has new security features. Tabbed browsing allows you to open several webpages within the same IE window. IE 7 and 8 tend to be compatible with more websites than Opera. However, IE 8.0 does have a couple of drawbacks. Users familiar with version 6.0 may take some time getting used to it, and might find some features more difficult to locate even after becoming accustomed to it. It also has high system requirements, making it impossible to use on many pre-2002 computers: 233MHz or faster processor, Windows XP/Vista/2003, and at least an 800x600 display resolution. The download time will be longer than most other browsers, as a 16.2MB file has to be downloaded before IE 8.0 will install.
Internet Explorer 6.0: While it lacks the new capabilities of IE 7.0, IE 6.0 may be preferable for some Internet users because of its lower system requirements: 66MHz or faster processor, Windows 98/ME/NT/2000/XP, and 16-32MB of memory. The vast majority of PCs being used today meet these requirements. It is also easier to learn to use for people familiar with older browsers and/or traditional Windows applications. Most websites will appear properly on IE 6, although a few major websites are strongly encouraging users to upgrade to IE 7. If you still have Windows 95, IE 5.5 is the latest version that will work.
Netscape Navigator 8.1.2: More popular in the past before Microsoft began including Internet Explorer with Windows, Netscape Navigator remains available and mildly popular but it is no longer being updated. According to the Netscape website, the capabilities of version 8.1.2 include an RSS feed reader, tabbed browsing, more security features, and access profiles for different users of the same computer. System requirements are somewhat high, but lower than those of IE 7.0: 233MHz or faster processor, 64MB+ memory, and Windows 98SE/ME/XP/2000. The download time for NN 8.1.2 will be long on a dial-up connection, as the installer size is 18 megabytes.
Firefox 3.6.12: A popular browser among Linux users, Firefox is also available for Windows and Mac OS X. It offers tabbed browsing, a built-in spell checker, the availability of numerous add-ons, and many other features. It requires Windows 2000, Mac OSX 10.4, equivalent Linux, or a more recent OS. The Windows version download size is 7.7 megabytes.
Firefox 2: This older version of Firefox downloads more quickly (5.7MB) and works on more computers. The Firefox website provides a downloadable add-on to the browser for using IRC-based chat services. System requirements are as follows: Windows 98 or newer, 233MHz or faster processor, and 64MB+ of memory/RAM. Linux requirements are similar. For Apple Macintosh computers, a G3 PowerPC processor, 128MB of memory, and OS X 10.2.x are required.
Opera 9.x, 10.63: Opera now has an appearance similar to Internet Explorer 7/8 with tabbed browsing, which it has offered for many years. There are built-in email and IRC chat features. The browser is available for many operating systems, including Mac OS, OS/2, BeOS, Linux, Windows 2000, and others. Opera is often recommended as the best graphical browser for older Windows computers; version 9.26 only requires Win95 and up, although a Pentium II or faster processor is recommended. The download size for the browser is also lower than average, at about 10MB for version 10.63 and four megabytes for 9.x.
Arachne: One of the few graphical web browsers for DOS, Arachne features a built-in dialer and WAV sound player. Additional add-on features are available, including an MPEG video player. It only requires 425K (0.4MB) of memory, MS-DOS 3.3+, and an 8086 or faster processor (a 386 is recommended; any Pentium should be fine). According to wikipedia.org, it is "currently the most advanced graphical web browser available for DOS." It can be used with dial-up or LAN internet access.
Safari 5: A browser primarily for Apple computers, Safari is designed to better support keyboard-only browsing, it boasts an extra-refined zoom function, and it offers a handy "minimum font size" setting. According to the Apple website, Safari "loads pages more quickly than any other Mac web browser." It is also available for Windows users.
TEXT-BASED WEB BROWSERS
Text-based browsers are generally more complicated to set up and some webpages will not display properly on them. However, they are faster and have low system requirements, so they are probably the best type of browser to use for accessing the internet on a DOS or pre-Pentium computer. An added benefit is that they do not load most types of advertising, including pop-ups.
NET-TAMER: This web browser allows almost any IBM-compatible computer with a modem to access the internet. In addition to webpages, it can use FTP sites and retrieve messages from your ISP-based email account. Net-Tamer is a complete Internet access package, including an ISP dialer. I recently connected to the internet on my old Tandy 286 (made in the early '90s) using this program, and I once accessed the internet with an 8088 using a 1200 bps (1.2kbps) modem with the same program, years ago. Net-Tamer is also one of the few browers which can fit on a single floppy disk, so a hard disk drive isn't even required. It is available in a few different versions, depending upon the type of computer you are using.
LYNX: A web browser for DOS, Windows, Macintosh, and Linux computers, Lynx is probably the most well-known text-based browser. According to a page on the Apple website, Lynx has become "quite popular" as a web browser for visually-handicapped people. Its system requirements are minimal; the DOS version should work on most computers with a 386, 486, or Pentium processor. I found Net-Tamer easier to set up than Lynx, but this probably varies from one computer and operating system to another; however, unlike Net-Tamer, Lynx can be used for Internet access through a local area network. It also seems to load and display a greater number of websites properly.