There is 4 different frequencies used by GSM handsets round the world. I
can't remember them offhand, but the UK uses 2, US uses 2 (one the same, one
different) and other parts of the world use others out of the 4.
A quad band phone should work anywhere in the world that has a GSM phone
network. Some places (Japan?) have limited or no GSM coverage (they use a
different system) so your quadband phone won't work.

A dual mode phone works with more than one technology, so (say) GSM and CDMA
(2G and 3G) or maybe GSM and WiFi(??)

Disclaimer: The above is based on a layman's understanding of the
technologies. Maybe a more qualified person could expand on, or maybe
correct my answer...

HTH

The only place I've heard the term "dual mode" used is in North America,
where it describes phones which support both CDMA and AMPS (1G analogue).
Quadband describes GSM phones, which are a whole other thing.  I guess
"dual mode" could also refer to phones which support both GSM and (3G) UMTS
(these may support 5, or even 6, bands), but I've not heard that used.

Dennis Ferguson