Both triband and quadband refer to the number of GSM bands that a GSM phone has. There are four GSM bands: 850, 900, 1800, and 1900. The most dominant bands in the United States are 850 and 1900, while the other two are used more in the rest of the world. A phone with three of those bands is called a triband phone, and a phone with all four is called a quadband phone. Triband phones can have any combination of those four bands, so you might want to check the phone's manufacturer or carrier to find out which three bands the phone supports.

As to which one you should choose for a world phone, both will fit your globe-trotting needs. Though a quadband is preferable just to ensure better connectivity, a triband will work well.