In a nutshell, broadband phone service uses the internet to route your incoming and outgoing calls. It's revolutionary and has now becoming a "real" alternative to traditional phone service providers.
Another technical term for broadband phone service is VOIP, or voice-over-internet-protocol.
The service requires a broadband internet connection (cable, DSL, T3, T1, etc) along with some type of phone adapter (each company provides their own compatible hardware).
It's now quite simple to install and setup your service, especially compared to your local phone company. Pretty much just plug and play... plug in the adapter to your cable modem and then to your computer, reboot and if all goes well, you can be making and receiving calls within minutes. Now that's pretty cool!
Your options are to either signup online with a service like Vonage (one of the leading providers) or visit your local computer store (or store that sells computers) and pickup a "retail" installation kit. Ordering online takes a few days to receive your adapter, but in my opinion worth the wait. You get a better adapter and if you use the link above, you also get a free month of phone service.
I personally use Vonage and have been really happy with the service, especially recently. It's taken some time for the industry and service providers to mature and work out the kinks, and I think it's now at a point where it's a viable alternative. So much so that I've cancelled my local phone service and use Vonage exclusively.
Let me know your experiences and any questions you might have.