This will be long, but reasonably complete.
Rifts and Greater Rifts are end game progression in D3. In all honesty, the game only begins at level 70, where the goal essentially becomes ‘how high can I make my numbers go’ and proving it by beating the hardest difficulty you can. GR’s are that mechanism by which you increase the difficulty beyond the selectable difficulty levels (normal through Torment XIII).
Rift and Greater Rifts can be opened at the obelisk in town manned by Orek. Rifts can be opened any time and GR’s from level 60, I think (or 70 if not). Both Rifts and Greater Rifts are essentially randomised dungeons that you progress through to ultimately reach a final boss - a guardian. When running a Rift/Greater Rift you will have a progress bar that fills as you do damage and kill mobs. The bar is also filled (much faster) by collecting orbs that are dropped by elite packs (blue and yellow mobs). To get the orbs off the blues, you have to kill them all, to get the orbs off the yellows, you only need kill the main yellow (unique) boss. Once the progress bar is filled, one of several guardians will spawn, which you then have to beat.
Rifts and Greater Rifts differ in a few ways, beyond the fact rifts can be opened at any level. Greater Rifts are the ones where you can increase the difficulty. When you open a normal rift, it’s difficulty is the same as the difficulty you selected at game creation, ie Torment X. Greater Rifts, on the other hand, you can specify a difficulty anywhere from 1 to as far as you have managed to get (plus a few). GR45 is equivalent to Torment X and the hardest difficulty setting in the game (Torment XIII) is the equivalent of GR60. Every 5 GR levels, roughly, monsters HP doubles and they hit harder.
You can only access Greater Rifts by spending Greater Rift keys, which are dropped by guardians of normal rifts. Normal rifts also drop loot as usual from trash mobs, elites, destructibles and of course, the guardian. Greater rifts only drop loot from the guardian at the end. Greater Rifts are also timed. To complete’ a Greater Rift, you need to enter, progress the meter and kill the guardian within 15 minutes. If you do so, you get an extra reward, which I’ll get to in a minute (legendary gems). It’s also worth mentioning that drop and XP rates continue to scale with GR’s, so the harder you can do, the better the drop rates and more XP you collect.
When people talk of farming, the are often talking of running normal rifts to get loot and GR keys. When people talk of pushing, they often mean running GR’s to see how far they can get (and to level legendary gems).
Now, the extra booty! When you complete a GR (can’t remember if it has to be in the allocated time), the Greater Rift Guardian will be guaranteed to drop a legendary gem. These are special gems designed to be slotted into jewelry slots - ring and amulets - and are game changers themselves, adding a variety of effects from raw damage, to mitigation, to AOE affects to self-proc bonuses, etc, etc. All legendary gems have two effects - a primary and a secondary. Legendary gems can and must be leveled to both improve the primary effect and ultimately unlock the secondary effect, which happens at level 25. There are currently around 22 different legendary gems and they will continue to drop from Greater Rift Guardians until you have one of each for that toon (which is all you need as you can’t stack their bonuses by slotting two of the same anyway).
In order to level legendary gems, you need to run more Greater Rifts. Upon successfully completing a GR in the allocated time, Urshi will appear at the Guardian’s corpse and offer you the opportunity to level your gems. If you don’t complete the GR in 15 minutes, no gem leveling for you! Generally, you have four chances to level a gem. If you die during the GR, that drops to three. Dying in a GR also adds a re-spawn time penalty, increasing with each death up to a max of 30sec per death. Dying repeatedly in GR’s is a sure way to fail them within the time limit. GR’s can also be empowered with gold when you create them to provide an additional gem leveling chance, which is unaffected by death. So, a perfect GR run will net you five total chances to level your gems. As long as you are running a GR 10 levels higher than a given gem level, the chance to upgrade that gem will be 100%. So to guarantee an upgrade on a level 20 gem, you need to run a level 30 GR. That chance drops to about 60% if the levels are equal (level 30 gem in level 30 GR) and diminish rapidly from there - ie it gets very hard to level gems that are higher than the GR’s you are completing. Important - if you wanted to level a level 20 gem a guaranteed 5 times, you would need to run a level 35 Greater Rift!
Legendary gems are a HUGE power booster, even (especially even), for fresh 70’s. And they continue to scale in power as you level them. It’s why sockets are mandatory on jewelry. Some gems define builds - the thorns gem for the thorns crusader, or the pet damage gem for any pet build, for example. Other are mainstays and will be in most of your builds (Trapped, which adds scaling flat damage boost to slowed enemies and provides an slow aura that self-procs at level 25, or Stricken, which incrementally adds more and more damage with each attack against the same enemy - perfect for taking down very high HP single targets like guardians that would otherwise be beyond your DPS within a given time limit).
And of course, because drop and XP rates scale with GR’s, the more you run and higher they are, the better you chances of getting loot upgrades and faster you accrue Paragon levels.
Finally, leveled legendary gems are used to augment ancient legendary gear using one of the cube recipes. You won’t use you main equipped gems for this, you will level off gems instead. Augmenting gear in this fashion adds a bonus to a given ancient legendary item - usually you will add main stat (str, int, etc) as your damage scales from main stat and ultimately, to progress, you need to do more damage.
So D3 endgame, when it comes to Rifts and Greater Rifts is - run rifts for loot and to collect keys, run GR’s to collect and level legendary gems, use leveled legendary gems to augment ancients, rinse, repeat until you get as high as you can or tire of the treadmill in any given season. That makes it sound more boring than it is! ;)
To complete most seasonal achievements you are going to need to get to around GR70. To get on the leaderboards, you are going to need to get to GR 90+. Non-seasonal toons with thousands of paragons are running up to GR110ish, but they are masochists. It is not really that hard to get to GR70, you can brute force it with enough paragon and and many, many documented builds will get there. Any set will do it. That is where I tend to tucker out, myself, as sometimes running very high GR’s becomes an exercise in precision and patience I don’t have the stomach for.
I have also mentioned before, but grouping makes this progression much, much faster - from power-leveling to sharing of drops, to just being carried for some paragon levels and blood shards for gambling. You will ultimately progress faster and get the drops you need by sharing your time with others.
