The halberd was originally designed as an all-purpose superweapon, hence the axe blade, the spear tip, and a hook on the rear side.
The hook was for dismounting mounted opponents, as Hugo mentioned.
The axe head was in fact, for chopping people. Like an axe. Hence "axe head".
The spear tip was for the spearwall formation as well as fighting other spear users (which were extremely prevalent at the time).
It was an extremely versatile weapon that evolved from farm tools (like many other medieval weapons), such as the voulge and the bill.
The problems with it was that it was hard to use properly and required a lot of training, especially compared to its predecessors and its successors, which were basically spears and very long spears. It was also very heavy and short ranged. Not to mention when you make a weapon that can do everything... you naturally end up with a weapon that specializes in nothing.
It was made redundant with the return of the pikewall formation, or basically a tight formation of very long spears. It's very strange and slightly ironic. Over 1300 years of military warfare technology, and we ended up using the exact same phalanx tactics used by Philip of Macedon in 330BC. It's made even more interesting by the fact that we returned to this formation as an answer to the use of gunpowder, line infantry, and the other military trends of the early gunpowder age. At that point, the halberd basically became a novelty.
Sorry for the detour. Please return to your regularly scheduled Skyrim now.