For mobile telephones there have been traditionally 4 frequency bands for GSM. These are known as 850, 900, 1800 and 1900. These numbers represent the frequency range (in MHz) that the band uses. As a rule of thumb, the Americas (except Brazil) use the 850 and 1900 bands, while the rest of the world use the 900 and 1800 bands. A comprehensive explanation is available in this wiki-page.

For WCDMA networks (aka 3G), Europe introduced the 2100 band, while in USA the so-called AWS band was created (that uses some spectrum in the 1700MHz and 2100MHz ranges) for T-Mobile to deploy its 3G network. As GSM networks age, GSM bands are used for WCDMA networks. A more detailed description of 3G bands is available here.

Due to the ubiquitous use of mobile services, the spectrum in the 6 above-mentioned bands is expected to be insufficient. This is especially true since 2G and 3G services will become legacy. That’s why that in the wake of LTE (aka 4G) more bands to accommodate the expected growth of mobile communications are being created. This was also possible due to the liberation of the spectrum used by analog services (analog TV mainly). Note that basically each country has the power to decide how to assign the freed spectrum. Due to this anarchy, as many as 34 new bands were created to date. See LTE bands.

The exponential growth in the number of bands is making it hard to manufacture radio hardware that can support all these bands. In fact, this was an issue even to give support to the 4 GSM bands: only high-end handsets will give you the luxury of the quad-band support, while feature phones will be market-specific. This was also due to certification issues: for example, to release a device that has to operate in the American market bands, it will certainly need an FCC stamp before it can be shipped.

As for LTE global devices, we can list the iPhone. Although it is not that obvious to final consumer, there are three variants of the iPhone to be able to cover all the LTE bands in use to date. Also, some countries are being quite deliberate to when it comes to decide what bands will be licensed for LTE use. An exception is Europe: all countries have already decided what bands will be used within the block, while in Asia and the Americas each country is going its own way, creating a lot of fragmentation on the bands to use. This will have a strong effect once roaming from one country to the other once LTE becomes state-of-the-art.