Europe uses GSM (Global System for Mobile Communications) phones.
T-Mobile, AT&T also use GSM phones in the US.
However they operate on different frequency bands. Cell phones in the US work at GSM 1900 or 850. Cell phones in Europe work on GSM 900 or 1800.
Both AT&T and T-mobile sell Quad Band phone (four bands) that will work in Europe also. (you may have to register for a special plan that allows International use).
With T-mobile you have to call and ask that world class roaming be added to your account. There is no charge unless you use it.

GSM phones use SIM (Subscriber Information Module) cards,a small chip that fits into your phone. This chip contains the cell phone number (so if you switch it to another phone, you still have the same number) and your account information. The SIM Card determines the network and phone number the cell phone uses.

You can use your existing SIM card with T-mobile in the UK.

Most cell phone providers in the US and Canada lock their cell phones (also in the United Kingdom). This means that you cannot change the SIM card, but if it is unlocked you can get a SIM card with a new phone number for the country you are in and don't have to buy a new phone.

Most countries in Europe have cell phone stores in the airport and in most towns and cities. You can purchase a GSM cell phone and SIM card when you are there. This option is cheaper, because the companies that sell these packages online add a surcharge to the cost of the SIM card.