The NFL draft is a system whereby each team gets to pick the college player that they want. The worst team of the previous year picks first and so on. On each round, the teams pick again. This allows the worst teams to improve their future position. The last player of the last round would be the player that almost nobody wanted. It is assumed he would not be very good or very valuable.
"Lowsman" is derived from "low" and "man". It is not a trophy that you would want to get if you were a football player being picked by professional teams.