There are two popular versions of Thor's Hammer. The one from Norse mythology, known as Mjölnir ("That which marks and pulverizes to dust").
Its power included "whatsoever might be before him, and the hammer would not fail; and if he threw it at anything, it would never miss, and never fly so far as not to return to his hand." And is"capable of leveling mountains."
The other version of the hammer is from the Marvel comic books and movies. The following refers to the hammer used by Thor in the movies.
According to the movies, the Asgardian technology is 1000 years ahead of us. To give you some idea of how advanced they are, just look at our civilization 1000 years ago:
Most of the population was involved in farming, with a village surrounded by two large and unfenced open fields. The village would have consisted of several small huts. They often housed the family's animals. There would be one or two rooms, with a loft for storage. The family lived in a single room.
There was no electricity, no running water (other than a river or stream), no medical facilities, nothing we now take for granted.
So first, a list of the powers of Thor's Hammer (not called Mjölnir in this case to distinguish it from mythology).
1) It can absorb and then discharge a lightning bolt.
2) When slammed to the ground, it can cause minor earthquakes.
3) Whenever it is spun by its leather strap, Thor can fly as fast and as high as needed.
4) It will always return to Thor's hand.
5) Only those deemed 'worthy' may pick up The Hammer.
1) The Hammer is very much like a battery - it will absorb an electrical charge, and then discharge it when necessary.
2 and 3) The Hammer contains a gravity / anti gravity generator. When traveling downward, the gravity generator half energizes, slamming The Hammer hard enough to cause the ground to shake. When it is spun fast enough, the anti-gravity kicks in and Thor is able to fly.
4) The Hammer returns to Thor's hand because of it recognizes Thor's DNA and "follows the scent" to return to him.
5) Although The Hammer may actually only weigh 100 - 150 pounds, if put on a scale, only those deemed 'worthy' will be able to pick it up. But who determines who is worthy? Odin, obviously. In the first 'Thor' movie it was Odin who stripped Thor of the power to wield The Hammer. So if Odin wanted to, he could deem anyone as worthy - Lady Sif, Heimdall, Jane Foster, or even Loki.