In nightmare and hell mode, the Area Level is basically used to determine the types of items (and their mods) that can be dropped from monsters, as well as changing to-hit calculations on a monster. It also also plays a role in determining the possible item types from chests and chest-like objects, in all difficulty levels.
For a useful Diablo 2 Expansion (1.10+) example, grand charms that appear with a +skillbranch mod can only appear on monsters that have been killed from The Great Marsh in act 3 in nightmare difficulty onwards. Technically, the +skillbranch mod can only appear on items of ilvl 50+, and the ilvl of an item is equal to the mlvl of the monster that dropped it, and the mlvl is equal to the area level that the monster was spawned in.
Via [1]:
Ilevel 50+ = plain skillers (plain mosters NM Great Marsh onwards)
ilevel 61+ = skillers with 31-35 life (plain monsters most of Act 5 NM onwards)
ilevel 77+ = skiller with 36-40 life (plain monsters Hell Lost City onwards; Pindle)
ilevel 91+ = skiller with 41-45 life (Hell Baal ilevel 99, Hell Nithy ilevel 95 and Hell Diablo ilevel 94)
Caveats: Named special monsters like Baal or Diablo have their own special mlvls, and drop items with an ilvl independent of the area level they spawn in as far as I remember. Champion monsters (monsters that have their name prefixed with fanatic, ghostly, etc) and regular unique monsters like "Devil Bender the Unclean" have an mlvl/drop items of ilvl that is +2 and +3 of the area level, respectively.
Via [2]