There are two major types of tri-band
GSM phones: European and Americas.
European tri-band phones support the 900, 1800, and 1900 frequency bands. They provide complete coverage in Europe, Africa, and Asia, and work on at least one network in most countries in the Americas.
Americas tri-band phones support the 800/850, 1800, and 1900 frequency bands. They provide complete coverage in the Americas, and will work on at least one network in most countries in Europe, Africa, and Asia.
As for quadband phones:
The four bands include the 850 and 1900 MHz bands - used in the Americas - and 900 / 1800, used in most other parts of the world.
Compared to a tri-band phone supporting the 900 / 1800 / 1900 bands, a quad-band phone adds support for
GSM 850 for full coverage in the Americas.
Compared to a tri-band phone supporting the 850 / 1800 / 1900 bands, a quad-band phone adds support for
GSM 900 for full coverage in Europe and Asia.