For ethernet, the standard link negotiation speeds are 10, 100 and 1000 megabits per second (faster standards are in the works, though). They're not directly related to how fast the data actually travels across your network (and of course the Internet), that's much more a function of the applications sending and receiving the data or the other systems competing for internet bandwidth between your computer and the target system.

Full-duplex is an attribute of the connection speed configured in your ethernet adapter's settings; in the list of selectable speeds you'll see each speed (10 and 100 in your case) listed twice, one showing half- and the other full-duplex. How you get to those settings varies by by operating system and version.

Another poster mentioned that auto-negotiation shouldn't be the problem, but in my travels I've run into it as an issue more often than I would have expected . . . this with an IBM/Lenovo ThinkPad T60 (now retired, but not that out-of-date) in data centers I would consider quite modern.