If this is new construction, chances are they'll deliver fiber to the prem, otherwise it'll be copper.
If it's fiber, the conversion happens at an "ONT" (like a NID - grey box on the side of your house). From there they can use either coax (HPNA) or Ethernet (Cat5e) to an "RG" (Residential Gateway). The RG then feeds your STBs, provides Wireless/802.11g, and, if you choose their VoIP voice service, this is the termination for your phones. There are two jacks in the RG. IF you do not choose the VoIP, then your POTS service is delivered via fiber, terminating in the ONT. Standard is also two lines, but more can be added.
If your service is delivered on copper, it's mostly the same. The service is VDSL (VDSL2 is "around the corner"), delivered to the RG. from there, everything is the same as delivery on fiber to the ONT.
The first box is usually a DVR, subsequent boxes are just STBs. With "Total Home DVR" all of the STBs can use the DVR for playback. All boxes are HD and have outputs for Composite, S-Video, Component, and HDMI. A typical installation permits "2+2" - 2 HD and 2 SD boxes showing or recording different programming at the same time. While those four are working, another four (maybe eight) can be watching something else via Total Home DVR.
U-Verse can be distributed in the home via either coax (HPNA) or Cat5/5e/6/6a. UTP is the preferred media, though if the installer sees decent coax, he'll try to use that ... I suggest you push for UTP. Some of the differential noise modes that show up on coax are difficult to filter well. These problems are not problems on Cat5/5e/6 (where it's more likely to be "common ode" noise and more easily dealt with.
I have it and like it a lot, I still have but will be giving up, Direct TV... which I've had since their Day One in the Chicago Market. I like DTV a lot too; U-Verse is giving me more features that I use for less money.
FWIW