Torghast is an interesting experience, especially as I've been playing Hades lately.
Torghast consists of multiple wings, each wing with multiple layers, and each layer with 6 floors. Two wings are open each week. You unlock layers by completing the previous layer (though once you've unlocked, say, Layer 3, you can access Layer 3 on any wing). Torghast's main reward is Soul Ash, which you need to craft Legendary equipment.
Each run of Torghast involves you going through the six floors in a layer. The layout of each floor is more or less random, with a vendor on the 3rd floor and a boss on the 6th floor. You have to beat the boss to complete the run and get the reward. As you progress you get random "anima powers" which boost or alter your abilities, or give you buffs. Most of the time, you get to select a power from up to 3 choices.
Torghast scales from solo to a group of five players. I've done runs solo, in a duo, and in a group of five. Elites and bosses have stacking buff which increases their damage, so there is a DPS check which you have to meet. So you have to balance anima powers between offense and defense.
It's pretty interesting because different builds can focus you on different powers. But so far, the boons don't do anything too drastic. You still play your basic rotation.
I recommend always choosing anima powers which will help you with the end boss. In my experience, the end boss is the most difficult part of a run. The closest I've come to failing was a run where I took too much defensive powers, and had real trouble meeting the damage check.
Another tip is, that at least when you are solo, it's possible to pull carefully and only get a small number of enemies at a time. For example, if two elites appear to be sparring, you can pull one of the elites without getting the other.
There's one wing where an assassin spawns when you are out of combat. I kept a mawrat in combat with me most of the time, and that stopped the assassin from spawning. Blizzard did nerf the assassins later in the week, though, so it might be easier to kill them.
Torghast is a pretty fun experience, and Blizzard has done a nice job of allowing you to farm or progress. If you just want to farm, you only have to do two layers each week to get the maximum rewards. The anima powers allow you to construct weird, broken, or amusing combinations, without breaking the rest of the game.