If your phone is unlocked (by default it is not unlocked if you are from the US or Canada, some providers allow you to call and will unlock for a small fee as long as you have been on contract for X amount of time, or there are places online that sell unlock codes), it allows you to put in a SIM card (identity card) from another cell provider.
This means that when you go international, you could go into 7-11 (or whatever convenience store) and buy a pay as you go plan & SIM card for a local provider, and put it in your phone. Then you use that phone number and data.
It is much cheaper if you are using it over the long term, especially so when data is involved. But the phone number is different, so people calling you would be long distance.
One thing to be careful of: Cell phones have certain Spectrums (or Bands) that they can communicate on. Different providers use different spectrums. Make sure your phone has the right ones for where you are going, as that may limit the providers you can use a SIM card from.
Oh, 2nd thing to be careful of: Not really possible with many Verizon phones, as they use different technologies and spectrums.