Roaming, in the cellular context, is the ability to operate your cellphone
over the network of a provider other than the one you subscribe to (the home
For example, I may subscribe to the Orange GSM service in the UK. I ask Orange to enable roaming on my phone. Now, assuming I travel to a country where there is a GSM operator with an agreement with Orange, I can use my phone on that operator's network. If I can wish, I can allow incoming calls - callers would use my normal cellphone number, and the call is automatically redirected.
Note, that there is usually a fee for enabling roaming, and a hefty charge for both incoming and outgoing calls.
If you intend to spend a lot of time in a particular country, it may be cheaper to buy a phone locally with a prepaid card, or if possible get a SIM card from the local company, and use that instead of your home SIM.
GSM is available throughout Europe and Asia.
Other types of phone (CDMA, TDMA etc) are also able to roam on compatible networks.