What are Greater Nephalem Rifts and how do they work?

Diablo 3's Patch 2.1.0 added a new feature called "Greater Nephalem Rifts". What are they, and how do they work?

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!

• Greater Nephalem Rifts are also timed. From entry, you have 15 minutes to kill the Rift Guardian to successfully complete the Greater Rift.
• Monsters inside Greater Nephalem Rifts drop no loot at all. Not even gold.
• No Treasure Goblins will spawn inside a Greater Rift.
• No shrines are present in Greater Rifts. One of each type of Pylon is present in every rift, though you may not see them.
• Players can self-resurrect at their corpse, however doing so adds a 5 second cooldown to the ability, stacking up to 30 seconds.
• You cannot teleport to a player that is inside a Greater Nephalem Rift, even if you are also inside the rift. Teleporting is enabled after the Greater Rift is completed.
• Interactable environmental items (bookcases, loose floor tiles, etc.) are nearly non-existent in Greater Rifts (I have found a few empty bookcases, but that is all so far), so items or abilities that trigger upon such interactions (like the Harrington Waistguard will be of reduced utility while within the Rift.
• You cannot change your skills or equipment whilst in a Greater Rift, so be sure to setup your build and equip the correct items before you start.

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.

There are several key points about the greater rifts to keep in mind. This is what I have learned so far from clearing up to level 33 solo.

Obtaining a Greater Rift Key

• Only a Nephalem Rift Guardian can drop a Greater Rift Key which is used to open a Greater Rift.
• The drop rates vary depending on the difficulty. Bagstone at DiabloFans has compiled a table:

Entering the Greater Rift

• Enter the rift at the obelisk.
• You can access any Greater Rift rank equal to or lower than the highest level you have completed. Completing a rift very quickly can unlock even higher ranks.
• No gear may be swapped while in the greater rift (or in town during the timer)
• The timer is always 15 minutes. If it expires, you still have to clear to 100% progress to spawn the Greater Rift Guardian.

Killing the Greater Rift Guardian

• No matter when the Guardian is killed, you will get a sizeable amount of loot, usually in the form of 1-3 legendaries.
• If you do not have all the legendary gems, the Guardian will always drop one or two. There are 19 greater rift gems (plus one from The Vault.)
• If the Guardian is killed within the 15 minute time limit, a vendor will show up and offer to upgrade your legendary gems. You get 3 attempts at leveling a gem (4 if you pay gold to empower the rift).

Upgrading a Gem

• Most gems can be leveled to a max of 100 (which isn't really going to happen). Some have a max of 50.
• Upgrading a gem has a percentage which is 100% if the level of greater rift was 10+ higher than the gem's level. 9 higher = 90%, 8 higher = 80%, 7 higher = 70%, 0-6 higher = 60%. If the gem's level is higher than the GR level, the rate goes down by one half every level, so 1 lower = 30%, 2 lower = 15%, etc. until you hit 1%. If the gem's level is 20+ higher than the GR level, the rate goes to 0%.
• At level 25, gems gain their listed secondary bonus