What are Greater Nephalem Rifts and how do they work?

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