A proxy server is a server which requests websites for you.
When you would like to request http://security.stackexchange.com, you don't contact stackexchange.com directly. You contact yourproxy.example.com instead, and ask it to get http://security.stackexchange.com for you. The proxy will then contact stackexchange.com with its own IP and relay the response back to you. From the point of view of stackexchange, it looks as if it is talking to the proxy server. It only learns the IP of the proxy, not yours.
Some proxy servers, however, add an additional
X-Forwarded-For header to their requests, which shows your original IP. This is optional, so a proxy advertised as a privacy tools should definitely not do this. Still, better be safe than sorry and validate that it doesn't.
But even when the proxy appears to anonymize you properly, keep in mind that you put a lot of trust on the proxy. The proxy can eavesdrop and modify your whole communication with the website, except when it is encrypted using HTTPS. Even when you use encryption, the proxy server can log your IP and which websites you requested when. Should your proxy receive an abuse complaint, they might reveal your actual IP to them, maybe even together with your detailed logs which can then be used against you.
That means you should think carefully about which proxy servers you are willing to trust. Many public proxy lists do not screen their submissions sufficiently. It already happened that hackers submit proxy servers to these lists which then skim passwords or infect users with malware.
When you still would like to use a proxy, you will find the proxy settings in the configuration options of your web browser. Firefox has them in the preferences menu under