Mobile phones work with different frequencies. Different cellular networks use different frequencies. Also, the standards are different in different countries. So cell phones have to be able to handle multiple frequencies so that they will be compatible with most networks. In India, we use 1800MHz (AirTel) and 900MHz (Hutch, etc.) so phones are at least dual band. The third frequency is usually 1900MHz which is used in the US. The US also uses 850MHz so a quad band phone has 850, 900, 1800, and 1900MHz bands. This enables it to be used almost everywhere in the world.
A Triband phone will have 900/1800/1900 or 850/1800/1900 frequencies. This means it will run on networks that require one of these frequencies. Helps while roaming so that you don't need a new cell phone.