What is Tor, and when to use it
Tor is a free network for browsing the web without handing over your IP address. The name comes from "The Onion Router", and the onion is a useful picture: your traffic is wrapped in several layers of encryption and passed through a chain of volunteer-run relays, each of which peels off one layer. No single relay sees both who you are and where you are going, so the site at the far end sees a connection from the Tor network rather than from you.
In practice you use it through the Tor Browser, a version of Firefox set up to route everything through Tor and to resist the tricks sites use to fingerprint you. Install it, open it, and you are browsing from an address that is not your own.
What Tor hides, and what it does not
Tor hides your IP address and rough location from the sites you visit, and it hides which sites you visit from your own network and provider. It does not make you invincible. Log into an account with your real name and you have identified yourself. Traffic leaving Tor to a plain http site can still be read by whoever runs that last relay, so stay on https. And it is slower than a normal connection, because your traffic is taking the long way around on purpose.
Tor and anonym.es are not the same tool
This is worth spelling out, because the two get confused. anonym.es is a dereferer: it hides the page a visitor came from by dropping the Referer header, and it does nothing about your IP address. Tor hides your IP address and location. One protects the page you were on, the other protects you. For real anonymity you often want both, each doing its own job.
If hiding your own address is the goal, start at the Tor Project. If you only need to keep your pages out of other sites' logs, a dereferer is enough on its own.