Did you know that virtually every website you visit logs your IP address?  This is the unique address that your ISP assigns to your computer each time you go online.  This therefore makes it possible for someone to track your internet activity!  There are ways around this, including anonymizing software, and anonymizing websites.  There are several options available, but as far as I can tell, the best methods are the following:

Anonymous Proxy Services

Anonymous proxy services allow you to anonymously connect to virtually any website as they mask your IP address. They is also extremely useful because you can use it to bypass a filter on a restrictive network so that you can access blocked sites (eg. webmail). There are many anonymizing websites available on the internet, but I have tried to list a few that I've found to be reliable. This is the most convenient way to surf anonymously or bypass network restrictions if you're using a public terminal (eg. at work) that you don't have software access to:

Below is a list of (in my opinion) the best anonymous proxy services.  They have little or no advertising and are recommended:

http://www.letmeby.com/
http://www.proxylord.com/
http://www.freetoview.net/
http://getwall.com/
http://www.proxydevil.com/
 

You should note, however, that using the above ad-free web-based proxy services do not guarantee strong anonymity - they simply mask your actual IP address from the sites you are connecting to.  Your IP will still be logged by the proxy site itself, so if strong anonymity is required, Tor is the better option...

Tor

If you want to surf anonymously at home (or on a computer that you are able/allowed to install software on), I can highly recommend Tor - a free software package released by the EFF to help you protect your privacy online.  You can download the Tor software package from:

http://tor.eff.org

This is a fairly simple piece of software that runs from your system tray. You need to configure your browser to use it as a proxy server (full instructions here for Windows users - other systems also supported) and it does the rest for you. Tor offers a good speed for browsing although download speeds are fairly low when getting large files. It's not perfect, but it works much better and more reliably than certain paid solutions such as Steganos Internet Anonym, and it protects your privacy as and when you want it to.  This provides strong anonymity since all your web traffic is "bounced" between a random number of random Tor nodes (other users), so it is very difficult (if not impossible) to track your movements on the web.  The speed of Tor allows you to surf relatively comfortably, but downloading large files is not only slow, but it puts an unnecessary strain on the Tor network and is therefore not recommended unless absolutely necessary.