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
Player:
I would like to hear more of the creation of our world.
Tyrael:
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.
This Blizzard Post on the upcoming patch Aptly describes the hidden benefit of the Nephalem Valor buff and the changes coming to it. (It's roughly 1/5th of the way down the post, "The Nephalem Difference")
Ahhhh screw it, here's the quote:
"It’s no secret that our goal for the end-game item hunt is players
hunting monsters packs, building to five stacks of Nephalem Valor, and
then killing a boss. While we’re seeing a lot of that occurring, what
we’re missing is people feeling like it’s worthwhile to continue
onward after killing a boss. To help hit that goal we’re lowering the
number of guaranteed Rare items on bosses when you have your full five
stacks of Nephalem Valor from two guaranteed Rares to one guaranteed
Rare (you still have a very good chance at multiple rares, it's just
no longer guaranteed). In exchange, all champion and rare packs will
now drop a bonus guaranteed Rare item when you have your full five
stacks of Nephalem Valor. The change benefits players with more
overall drops, and a reason to push to continue progressing."
Best Answer
A Greater Nephalem Rift is a timed rift whose difficulty is independent of the current game's difficulty setting. Monsters inside Greater Nephalem Rifts drop no loot except for the Rift Guardian. The time limit for any Greater Nephalem Rift is 15 minutes.
Note that Patch 2.3 greatly overhauled (read: simplified) the Greater Rift feature.
Ok, so how do I open a Greater Nephalem Rift?
Firstly you need to complete one or more standard Nephalem Rifts to hunt for a Greater Rift Keystone:
These keystones are a random drop from Rift Guardians. If you are playing in a party everyone in the party will receive one if one drops. Once you have a Greater Rift Keystone, activate the Obelisk in town and select Greater:
Once you have selected the Greater Rift option, you will need to set the difficulty in the dropdown as shown above. Clicking Accept will prompt each eligible party member to enter the rift. Once everyone has accepted the invitation, everyone is immediately teleported to the start of the rift and stripped of all buffs.
Great! Is there anything I should know before I start?
Yes!
As with regular Nephalem Rifts, every non-resurrected monster that your party kills contributes toward spawning the Rift Guardian. Packs of Champions (blue) and Rare (yellow) monsters will drop purple Progress Orbs when defeated (with Champions, only the final Champion monster will drop Progress Orbs. With Rares, only the Rare itself drops Progress Orbs, minions don't).
This is a Progress Orb:
Each champion or rare pack of monsters will drop 3 of these orbs which must be "picked up". Each orb grabbed will move the progress bar 1% so it is important to get all 3 each time a pack is killed.
Unlike regular Nephalem Rifts, time is of the essence. Ideally your build would reflect that, giving you as much mobility as possible without sacrificing too much of your ability to kill things quickly. If a particular type of monster is taking a long time to kill, ignore it and move on. You can see your current progress and the time remaining on the progress bar on the right:
Once the progress bar is full, all monsters immediately de-spawn and the Rift Guardian spawns. Rift Guardians are the same as those found in normal Nephalem Rifts.
Killing the Rift Guardian will award a sizeable quantity of loot, with at least one legendary Gem (note: legendary Gems will not drop if the character has the same gem either in their stash, or in their inventory; this means that you will get no legendary gem drops if you already have one of each... however, putting a gem into the inventory of another character allows you to obtain multiples).
If you killed the Rift Guardian before the timer expired an NPC named Urshi will then spawn and allow you to apply a total of 3 attempted upgrades spread over any legendary Gems that you have (they can be in your stash so there is no need to carry them with you). Note that these upgrades have a chance to succeed based upon the level of the current Greater Rift compared to the current level of the gem you are seeking to upgrade:
Chart courtesy of Diablo Wiki
Once you have chosen, the rift is complete and you can safely exit the rift.