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 mutual 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 may not be a default feature (it involves 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). This used to be an issue, especially for US subscribers to Verizon, US Cellular and Sprint which all use CDMA - many countries have GSM networks only. Luckily, most mobile devices manufactured in the last few years are capable of connecting to both types of networks with their numerous frequency ranges. Double checking with your wireless provider wouldn't hurt, but chances are, if you are using a modern phone you will be able to use it anywhere in the world.
Note that you can not dial the numbers in your Contact list 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 locals dial without adding any international codes. In other words, while in another country your cell phone becomes a local phone for all outgoing calls.
One other thing you can do, especially if you are a frequent international traveler, is to store all your contacts using +country code - national subscriber number. This way, when you are abroad, your international calls will connect without adding any extra digits, whereas dialing domestic calls will also work this way in most cases. For instance, US numbers would be stored using +1 ??? ??? ????, UK numbers +44 ?? ???? ???? and so on.
While traveling abroad your cell phone number does not change for incoming calls, no matter where they come from. In other words, people from your home country will just dial your number without adding any extra international codes whereas everyone else, including your travel companions, will have to use your full international number including your home country code.
You do, the person traveling to another country. Make sure you inquire about rates - they are not low.