The Alpha Stag is an epic and rare beast. It is bigger than all other Deer in the game, endures a lot more damage and he is faster than other deer. It is therefore easier to lose him during a chase. He is easily mistaken for a regular brown deer if it's hard to spot the size-difference between the two.
The Alpha, contrary to its subjective relatives, do not make any appearance unless the player lures it out with Deer Scent. There is a bigger chance of making a Scent-Lured Deer appear instead of the Alpha. There is a book named Hunting The Alpha found inside Hunting Shacks that explains how to best find and hunt him down.