Greater Rifts are a new type of Rifts which will be added with Patch 2.1.
There are 100 difficulty levels of Greater Rifts, with level 25 being on par with T6.
Each complete Greater Rift run consists of multiple stages:
First, a Keystone of Trials needs to be acquired from a regular Level 70 Nephalem Rift Guardian in Torment 1 or higher.
This item can be used to open a Portal to the Realm of Trials which tests your strength by sending multiple waves of enemies at you which you have to defeat in a given time limit. Their power depends on the difficulty you're on when opening it.
You will then get a Greater Rift Keystone
of a certain rank, according to your performance in the trial.
This ranked Keystone is then used to open a Greater Rift of the appropriate level. The mosters' strength in there only depends on the Greater Rift level and no longer on the game difficulty.
If you manage to summon and defeat the Greater Rift's Guardian within 15 minutes you will get the opportunity to get a new Greater Rift Keystone of a higher level, depending on how long it took you.
You can continue playing harder and harder Greater Rifts until you can no longer beat one within 15 minutes.
Every time you beat a Greater Rift Guardian, you will summon Urshi, a Nephalem Ghost.
At Urshi you can choose to get a new, higher level Greater Rift Keystone. This option is only available if you beat the current Greater Rift in less than 15 minutes.
Alternatively, you can choose to attempt upgrades of your Legendary Gems at which point your Greater Rift run will end, not matter if you beat the timer. Read more on Legendary Gems and their upgrading mechanics here.
You may only choose one of these two options every time you finish a Greater Rift.
Opening a Greater Rift requires every party Member to use a Keystone of the appropriate or higher level.
Death in a Greater Rift or the Realm of Trials in Hardcore mode will be permanent.
Your characters' best Greater Rifts attempts will be saved on the Leaderboards, so you can compare with your friends, clan mates or the entire server.
Enemies in Greater Rifts will drop no loot and there will be very few clickable objects. The Greater Rift Guardian will instead drop a large amount of items with a very high chance of Legendaries.
Health and Nephalem Glory Globes will still drop though.
Champion and Elite packs in Greater Rifts will also drop new purple progress Globes which boost the Progress Bar when picked up, getting you closer to summoning the Greater Rift Guardian.
The Progress Bar in Greater Rifts will show both, the current progress and the timer, allowing you to judge your speed at a single glance.
Keep in mind that there are changes coming to regular Nephalem Rifts as well and that they also apply to Greater Rifts.