3G is a set of standards for the wireless transmission of data and voice. To laymen, it basically is a "pretty fast way to transmit data over a cellular network." So when someone talks about a "3G network," they're probably talking about a network set up by a cellphone provider (AT&T, Sprint, Verizon, Vodefone, etc, etc, etc) to provide their users with data connections on their cell phones.
Older, "2G" networks are generally slower and do not often allow simultaneous data and voice connections (i.e., you can't talk on your phone and surf the internet on your phone at the same time).
3G networks (when combined with phones that support 3G) generally provide data connections from 300kbits to 1,400kbits. Compare that to a dial-up connection (usually 30kbits to 50kbits), a DSL connection (usually 1,500kbits to 10,000kbits), or a Cable modem connection (usually 3,000kbits to 20,000kbits) and you'll see it's not necessarily "fast," but it's certainly "fast enough" for a cell phone.
By the way, AT&T is currently working on deploying a new set of hardware for their 3G networks that will provide speeds up to 7,200kbits, so 3G is getting to be more like a DSL or Cable modem when it comes to speed.
Not sure if that's what you were looking for, but hopefully your answer was in there somewhere. If not, check the Wikipedia article (below) for more information.