Proxy servers are used to try and hide your IP address. For example, you're at work or school and some websites are banned, so you connect to a proxy who then passes your page requests to the page you really want. Anothe rexample, you're banned for some reason from accessing a web server, by using a proxy they see the proxy IP address and not yours.
The problem is that there are ways of tracking you even through a proxy and lots of schools and business networks keep lists of proxy servers and will not let you connect to them.
Another problem is that because you're purposely connecting to a proxy all your internet traffic passes through it - whch means, if they want to, everything you send or receive can be seen by the owner of that computer.
Yet another problem with them is that can slow your internet conenction down.
More about proxy servers at http://en.wikipedia.org/wiki/Proxy_server
Lists of proxy servers at http://www.publicproxyservers.com/
EDIT: If you're paranoid about being hacked then using a proxy server is NOT a good idea. You have no idea who runs it and what they are doing to the information going through it. Absolutely anyone can run a proxy server - http://www.google.com/search?q=proxy+server+software
Take a look at that list of proxy servers, most come and go - good in the sense they may protect your IP address and need to change frequently to stop being blocked but very bad a security sense. Anyone can set up a proxy server, run it for a week or so, disappear with anything interesting they found, then start another.