A side of the Hammer is inscribed with: "Whosoever holds this hammer, if he be worthy, shall possess the power of Thor."
The answer varies by interpretation of the writer. Different writers have chosen to interpret it in a variety of ways. Some transfer the powers, some just allowed Mjolnir to be lifted.**
We have seen people become Thor (the crippled Donald Blake became Thor). This was later rewritten to be a mortal body created by Odin to teach Thor humility.
Eric Masterson was an EMT who was merged with the power of Thor by Odin when Eric was mortally injured as a bystander near a battle where Thor was involved. Later when Thor was thought to have killed Loki, Masterson was deemed worthy and retained the power of Thor and Mjolnir becoming Thor and even serving as an Avenger.
Thor later is released from banishment and Masterson gets his own hammer and codename: Thunderstrike. Thunderstrike is later killed in battle destroying the Bloodaxe.
We have seen the acquisition of Thor-like power such as Beta-Ray Bill. Beta-Ray later gets his own hammer Storm Breaker. Beta-Ray was an alien who acquired the full powers of Thor and still retains them today.
We have seen people simply lift the hammer of Thor and no powers are transferred. Captain America was one such person.
When you first spoke to me about your problems, I had doubts...about you. They were quickly erased...when you lifted Mjolnir...for only a man or god worthy -- pure of heart and noble of mind -- could have done so! ... A sacred bond unites all those who have e'er been privileged to wield Mjolnir! A bond which stretches far into infinity! I salute you, Captain! --Thor Vol 1, Issue #390
Others lifting it in canon include: Erik Masterson, Beta-Ray Bill, Odin, Bor (Thor's grandfather) and Buri/Tiwaz, Thor's great-grandfather.
Several sentient machines have also lifted it (non-sentient machines cannot): Gabriel Lan, the Airwalker and herald of Galactus, Zarrko the Tomorrow Man's mining robot, and the Awesome Android by imitating Thor's abilities and worthy nature.
Thor #340 reveals that when Thor loses the hammer for 60 seconds, he reverts to mortal form. There have been several besides Thor who have been able to wield Mjolnir, Captain America for example (Thor #390).
They don't become Thor, but they wield the power of Thor, which is Mjolnir the epic hammer. Note that during times that others wield Thor, by the same enchantment previously noted, Thor loses his "Thor powers".
To clarify, when Captain America got Thor's hammer, he didn't get Thor's strength, he only had the abilities of Mjolnir. During this time Thor's strength was gone, but as soon as Cap' tossed the hammer back to him, his strength came back and he mashed his opponent through an island (that had water under it...).