This is a little basic, but:

In the US, the four major companies use one of two technologies:

CDMA: Verizon &Sprint

GSM: AT&T and T-Mobile

The base difference is that phones on a GSM network use a removable "soft card" to identify the phone to the network,

A CDMA phone identifies to the network with internal circuitry,

This means that a CDMA phone can't be used on another network.

With an UNLOCKED* GSM phone, by temporarily swapping out the U.S. SIM card and sliding in a SIM card obtained from a European (for instance) provider, your phone will function on the European network. Your contacts and apps are intact, since they typically reside in the phone's built-in memory.

When you get back home, swap the SIMS in reverse to get the GSM phone back onto the US network.

Example of SIM card: