The second and more effective way is by using a Tor client. The Tor client is a great way to make your Internet use anonymous and hide you IP. But, it is impossible to completely hide your IP address (see Warnings below). Visit the Tor website (google tor) and download the version of Tor appropriate to your system. There are Mac, Windows and Linux versions all available. Install Tor and restart your computer. Launch the Vidalia application, which is part of the installation, to start running Tor.
Tor basically re-routes your connection through other computers, and servers around the world. It's like a proxy but sends it through multiple servers so it's very, very hard to trace.
Configure your browser (preferably Firefox) to work with Tor. The installer will automatically install the Tor button to Firefox, which allows you to toggle Tor "on" and "off."
Disable browser plugins such as Java and Flash that might bypass Tor and reveal your IP address without you knowing. View what plugins you have installed by typing "about:plugins" (without quotes) in your address bar. To disable a plugin, open the Add-ons list under "Tools," then disable the plugin of your choice.
Use a web-based proxy service instead of Tor if you're only interested in web-based anonymity. Eatproxy maintains a list of popular and new proxy services to anonymize web browsing.