Typically, a phone that supports a specific technology in three different radio frequency bands.
For example, some phones support tri-band WCDMA, where the three bands are 850, 1700, and 1900 (all American bands.) Another example of tri-band WCDMA would be 850/1900/2100 (2100 is primarily a European/Asian band.)
Another example is tri-band LTE phones that support 850/1900/2500, and therefore fully support Sprint's LTE network as of late 2013.
Before 3G, tri-band meant a GSM phone that supported three of the four major GSM frequency bands, and would therefore work in most parts of the world, making it a "world phone". (Of the four major GSM frequency bands, 850 and 1900 are the two bands used in North America. 900 and 1800 are the most common bands in other parts of the world.)
GSM phones that supported fewer bands (dual-band) were generally not world phones.
Tri-band GSM phones were eventually replaced by quad-band GSM phones, once newer antenna and radio technology made it easy and affordable for nearly all GSM phones to be quad-band.
Tri-band is not be confused with tri-mode. Tri-mode usually referred to a CDMA phone, while tri-band usually referred to GSM phones.