All about He-Man
Early mini-comics
In the illustrated books released with the first series of toys and drawn by Mark Texeira, He-Man is a barbarian from an Eternian tribe.
The planet's inhabitants are dealing with the aftermath of the Great Wars, which devastated the civilizations that once ruled supreme over all lesser beings. The Wars left behind advanced machinery and weaponry known only to select people.
An early incarnation of the Sorceress of Castle Grayskull gives He-Man some of these weapons, and he sets out to defend the secrets of Castle Grayskull from the evil villain Skeletor.

He-Man possesses one half of the Power Sword; the second half is possessed by Skeletor, who uses it as his main weapon. When joined, the two halves of the Power Sword will provide the key to Castle Grayskull. This is why the two figures' swords could combine into one sword, when the action figures were initially released.
In one early illustrated story, He-Man and Skeletor actually united their two Power Sword halves to form the true Power Sword in order to defeat a common enemy.
Origins of He-Man
By the time the animated series was being developed, He-Man's origins had been revised as follows: his true identity is Prince Adam of Eternia, son of King Randor and Queen Marlena who rule the planet of Eternia along with Adam and the rest of their inner circle.
Prince Adam is shown here, from the 1980's TV animation:
The Sorceress of Castle Grayskull endows Prince Adam with the power to transform into He-Man, which Adam does by raising his Power Sword and proclaiming:
"By the power of Grayskull... I have the power!"

He-Man Companions
Prince Adam's pet is a cowardly green tiger named Cringer. When Adam becomes He-Man, he similarly transforms Cringer into a brave armoured green tiger named Battle Cat by pointing his sword at him - an ability Adam discovered accidentally the first time he turned into He-Man.
Cringer, naturally, cowered in fear at seeing what Adam had become and while reassuring him that nothing had really changed, Adam pointed the sword at Cringer, which sent a bolt of energy toward the tiger and transformed him.

Battle Cat serves as He-Man's steed and fierce fighting companion and here they both are from the 1980's:

More He-Man Characters
Adam is friendly with the beautiful and strong-willed Teela, the adoptive daughter of his mentor Duncan. Adam and Teela grew up together and now, as Captain of the Guard, Teela is entrusted to protect the prince. She often sees Adam as lazy and cowardly because she is unaware of his alternate identity as He-Man.
Teela is shown below as she appeared in the 1980's series:

Duncan, also known as Man-At-Arms, is He-Man's closest companion and the Eternian royal family's innovator of technology and weapons. In many episodes, Man-At-Arms unveils new and fantastic weapons or devices that help He-Man and his friends.
Duncan is pictured here, again from 1983:
Castle Grayskull is the source of He-Man's powers. Inside the Castle lives the Sorceress, who grants Prince Adam his transformative abilities, and communicates telepathically with He-Man.

To protect his family, He-Man keeps his double identity secret, sharing the knowledge only with Orko, Man-At-Arms and the Sorceress.
Here's Orko (1983):
The spin-off cartoon series She-Ra: Princess of Power later revealed that Adam has a twin sister, the Princess Adora, a leader in The Great Rebellion against Hordak on the planet Etheria. Adora, like Adam, is given the gift of the power of Grayskull and has her own sword which she uses to transform into She-Ra, Princess of Power.

He-Man also makes a number of appearances in the She-Ra: Princess of Power television series.
He-Man's chief adversary is Skeletor, a blue-skinned sorcerer with a skull for a head, wearing a cowl. He is skilled in black magic as well as all forms of combat. Though his origin is mysterious, and the cartoon describes him only as a "demon from another dimension," a tie-in comic implies that Skeletor's true identity is Prince Keldor, older brother of King Randor, thus making him He-Man's uncle.

It is revealed in the animated motion picture He-Man and She-Ra: The Secret of the Sword that Skeletor was Hordak's right-hand man up until his capture, and supposed release. Skeletor is accompanied by a group of henchmen who aid with his evil schemes.