There are both "good" and "bad" reasons to want to surf the Internet anonymously. We can all guess at least one of the reasons one might want to keep their surfing private, and it's not just because you want to be an evil hacker.
But other reasons can include wanting to hide one's identity when posting information or comments as a critic or whistleblower. It could just be the belief that we all have a right to privacy even when surfing online. In today's "Homeland Security" world, it is not just the "bad guys" who do not want the government prying into their private and personal business. Many people see it as a basic fundamental right: the right to privacy.
You may want to hide your identity from the sites you visit or simply hide your surfing from local eyes. It does not matter. Since the beginning of the Internet, demand and opportunities for anonymous web browsing have reached.
Anonymous Internet surfing services generally work this way: you sign into the service (called a proxy service) and tell it the site you want to visit. It – instead of you – visits the site, then shows the page to you. What the visited website gets is the info (such as IP address, browser type, country) about the service, not you. They can strip cookies, scripts, images and so forth so that a website's tracking devices do not make it as far as your computer.
Free services tend not to encrypt your information, however, so there can be a track left behind that others could see in the service's own log files, for example. Should they be subpoenaed and turn over their files, you could be traced. This may not be the type of protection you are seeking.
And, of course, a record is left on your own computer unless you have taken steps to prevent that from happening as well, such as turning off cookies and search history. There are numerous privacy software suites available that can assist you with this task of erasing footprints and evidence from your computer.
Now … you should know that anonymous surfing is not 100% guaranteed. It can not be. Even so-called "bulletproof" anonymous options can be broken if what you did was considered bad enough to spend the resources on tracking you down. Consider that, if a human being can create a software program, another human being can hack it … and will have fun doing so!
Your best bet is to do your homework. Research Internet privacy software available and read reviews of popular proxy servers to find the best fit for your needs.