On the internet nobody knows you are a dog
Unfortunately it's only true in cartoons! Basically you are leaving a surprisingly easy trail of the websites you visit. Visit Test anonymity if you want to find what web servers can know about you. A determined person can find out about the websites you browsed, what you did on each of them etc.
There are some commercial services like anonymizer that insert a random proxy between you and the destination web server. There are also a number of HTTP/Socks proxies that you can use. But then all of your traffic is subject to monitoring by these people.
Freenet project takes anonymity to other extreme and you can access content that you may not access otherwise, and also provides anti sensorship / banning features. But it has always been very slow, prone to protocol changes. (i.e. sites working the previous day do not work the next day because of release of new protocol and peer software).
Tor project takes another approach for this. The endpoints are still the same, but all your packets are routed using random combinations of tor routers. The routing technology is called onion routing where the encryption is only between hops in the route and none of the intermediate hops know either the contents of the packet or the sender. There is a provision for hidden services(any TCP protocol), which are not accessible from regular internet, which comes close to achieving what freenet does. I have been using tor for some time now and noticing some things:
* The performance is improving a great deal (as more and more tor nodes are commissioned, it will yield better performance)
* You can get routed through completely different continent, so going to google might open their german page (because they send you german page if they detect your IP address is from germany)
* This service might be easily abused by spammers who will definitely want to route spam through tor, child pornographer who can host "hidden services", illegal content downloaders. (Though I believe many tor nodes block SMTP and peer-to-peer traffic). I guess there is a price to be paid for "really free speech"