These two refer to hockey leagues. The AHL refers to American Hockey League while NHL refers to National Hockey League. There are some hockey players that usually start playing for the AHL. If they are considered to be really good and they have a future in playing hockey, they may be promoted to play for the NHL.
One of the main differences is the salaries that the players get, those who are part of the NHL will get higher salaries as compared to those who are playing in the AHL. Players who are already in the NHL are considered to have better hockey-playing skills as compared to those who are in the AHL.