At the end of the game, humanity will earn a seat at the Council… either alongside the original Council that has been causing you grief all game, or alongside a new Council, depending on your choice earlier regarding the dispensation of Alliance naval resources. After your heroics, your nomination carries significant enough weight to determine who the first human councilor will be:
Choice: Nominate Anderson or Udina
Depending on your actions throughout the game (whether you’ve been more Paragon or Renegade) Udina will have a different impression of humanity’s opportunity, either demanding parity with the Council races, or dominion over them. This change of tone has little bearing on the choice at hand, however, and you can nominate whomever you want: steadfast, competent, loyal Captain Anderson, around whom your recent career has formed, or Udina, a career politician.
In Mass Effect 2, Udina will prove considerably more acrimonious to Shepard than Anderson, although you’ll frankly have little interaction with the Council either way. Anderson will push to get you reinstated as a Spectre, regardless of your earlier choice regarding the Council, while Udina will oppose it - successfully, if he’s sitting on a new Council. This has little impact aside from lore.
By Mass Effect 3 Anderson has grown weary of politics, and Udina will take his place on the Council, if he wasn’t assigned there already.