Nephalem are the children of mixed parentage - angels and demons who created and fled to Sanctuary to avoid the perpetual war between heaven and hell. "Sanctuary" is the primary place where the Diablo games take place, and all of the humans are descendants of these first Nephalem. True nephalem are able to tap into their supernatural powers. Not every "human" in Sanctuary can tap his or her supernatural powers, but the heroes in the Diablo series can.
The paper manual that comes with the boxed copy of the game has a bit of backstory/explanation:
In the midst of this battle, a group of rebellious angels and demons sought refuge. [...] The children of these renegades were known as the nephalem, and they possessed a power greater than their parents could have imagined. [...] As generations passed, the nephalem forgot their heritage. They called themselves human and built their cities, nations, and armies without knowing their true potential.
(it's continued on another page, but it's unrelated to the current question.)
Tyrael explains this to you, if you ask him about "The Creation of the World" during Act 3, Machines of War:
The Creation of the World
I would like to hear more of the creation of our world.
To escape the Eternal Conflict, a group of renegade angels and demons came together and created a hidden paradise for themselves. They called this world Sanctuary.
Eventually, they commingled. They gave birth to mankind. That is why you have both good and evil within you... and the power of both realms as your birthright.
The power of your nephalem ancestors was so terrifying to their parents that they tuned the Worldstone to weaken them. They knew they had loosed something... unimaginable upon creation.
Some additional lore is available in act 4.
Auriel also explains this in a 4-part lore series titled "The Creation of Sanctuary"
After he gained a following, Inarius stole the sacred Worldstone and used it to create and conceal a world he called Sanctuary. Demons and angels alike fled there, and some of them—including Inarius and Lilith—fell in love. Many of these unions resulted in children. They were called the nephalem.
Inarius was alarmed when he realized that the nephalem had the potential to surpass both angels and demons in power. He wished to limit their abilities, but Lilith demanded that they become her army. Their escalating conflict led to Sanctuary's discovery by the demon lords and the Angiris Council.
Itherael mentions the Nephalim in "The Destiny of the Nephalim"
Angels and demons are bound to their natures. Demons are given to chaos and deceit; angels to truth and order. [...] The nephalem alone have choice. [...] These creatures dwell outside the order... They cannot fathom the freedom they possess.