If you have a cell phone you can take it with you when traveling to another country and you can place or receive calls without changing its SIM card.
Cell phone carriers from most countries have roaming agreements that basically allow you to use their wireless network when traveling abroad. Generally you do not need to do anything - the foreign carrier will take over so to speak and you will see another carrier name on your cell phone's screen once you are in another country. What you do have to do is call your wireless carrier before leaving your country and activate your roaming service. This is not a default feature (it implies extra costs).
There are two main technologies used with cellular phones: GSM and CDMA. A CDMA phone won't work on a GSM network (the opposite is also true). So, before traveling call your wireless carrier for specific info.
Note that you can not dial the numbers in your phone agenda as they are - you must add/drop some digits. Use these international dialing tools to find the codes you must dial.
To make local/national calls within the country you are in just dial the numbers they dial without adding any international codes.
People from home can call you AS IF YOU WERE HOME. In other words they do not need to add any digits to the number they usually dial when calling you.
You do, the person traveling to another country. Make sure you inquire about rates - they are not low.