IMO there's no good reason to need to mask your IP. Ignoring that, proxies are the way to go.
Proxies are a relay server. You connect to them, and through them connect to your desired website/Internet source. They can either be used as a gateway (I.e. You connect to a website then, using the tools on the website, browse the Internet from that window), or as an actual relay (this requires some slightly more complicated configuration in your web browser/download manager settings).
There are several "levels" of proxies which offer increasing levels of security. The first level is a basic redirect and won't actually hide your IP (think of call redirecting - you can usually still see the call number on the phone display). The highest level completely erases the e-trail of your presence so you're mostly invisible on the Internet. Hackers might even go one step further and use "zombie" terminals to scatter any traces to multiple connections, making it harder to trace back (this appeared in a certain James Bond movie).
Due to my principles, I won't name any. If you really need to use one, google up a guide. This post is for educational purposes only, and I take no responsibility if you screw something up and find yourself temporarily unable to use the Internet (which is what happens with a badly configured proxie).
As a side note, many proxies will log whatever activities you're doing and some even sell that data to advertisement companies. Do not use active proxies when connecting to important websites like a bank site.