A SIM (Subscriber Identity Module) card allows portability when changing cell phones. All of the specific account information, plus the subscriber account information and secret key code, are stored on this card on some models of phones. If someone accidentally breaks their phone, say by dropping it into a punch bowl full of alligators, they don't have to worry about their data, including their contact list.
Some phone companies do not use SIM cards, which makes their phones non-portable without cloning equipment. Verizon Wireless is one such company. AT&T, on the other hand, has phones that use SIM cards. If you were to go to another country, for example from the United States to England, you would have to get a local SIM card and just pop it in your current cell phone.
The SIM cards are small (around a half-inch in size), and are based on smart card technology. To install one, open the phone, find the SIM slot (which is sometimes under the battery), and slide the card in until it clicks. Some slots have a metal retaining clip. Make sure you line the SIM up correctly so the connectors line up with the contacts on the card. The SIM cards have a notch so they can't be installed incorrectly, but some people will try and force it in anyway.