What kind of cable connects from the wall to my wireless U-Verse router?

I recently moved from a house that had Gigabit (fiber) to one that only has 25Mb. The guy ran the line from pole in the alley, to my house. On the inside, he placed a box that connects a white wire to the AT&T U-Verse wireless router (residential gateway?). Anyway, the cable is very long. I would like to clean that up by replacing it with a shorter one. What kind of cable should I get, and were? If I unplug that cable to replace it, do I need to power cycle the router, or will it recover when I re-connect? Also, once they get fiber in my area and I opt for the Gigabit,  will that same cable be used? I assume the line coming to the house will be fiber. Thanks in advance.