The Labyrinth is a randomly generated dungeon filled with traps, puzzles and monsters. Completing the Labyrinth for the first time unlocks a character's Ascendancy class and points for the powerful Ascendancy skill tree. It also offers equipment enchantments, several unique poe items, and competitive leaderboards.
To complete the Labyrinth on each difficulty, you'll first need to do the Trials of Ascendancy. These are found in the less-used areas of the main story, and each one focuses on one of the types of trap you will encounter in the Labyrinth. I'm not sure where the Trials will be from Act 5 onwards, but for the first Labyrinth run they are located in: The Lower Prison (Act 1), The Crypt Level 1 (Act 2), The Chamber of Sins Level 2 (Act 2), The Crematorium (Act 3), The Catacombs (Act 3), The Hedge Maze (Act 3).
The Labyrinth itself can be tricky, but with a good enough build you should be able to do it. Avoid the traps, and you generally want to only go through rooms that you have to in order to reach the next Aspirant's Trial. In the Aspirant's Trials, you will face off against Izaro, who can be pretty hard. Can't really say much here other than that you should probably run in circles to mitigate how many times he hits you. The form where he dual-wields swords is particularly dangerous.
Once you start doing maps, you will begin seeing a new set of Trials of Ascendancy. These Trials are much harder but you now only need to do them once per account per league. At the end, it should drop you an Offering to the Goddess, which when put in the map device will open up the Uber Labyrinth. Be careful, as this fight is very difficult and you should probably be running Tier 10-11 maps easily to even hope to complete it solo.
Farming the Uber Labyrinth can actually make you some pretty great money. If you bring an item such as a Rat's Nest or a similarly well-used unique helmet, and enchant it at the end of every run, you'll eventually hit a sought after one that can be really valuable. For example, a friend recently told me he bought a Rat's Nest with Tornado Shot Critical Strike Chance for 17 exalt. Rat's nest is a couple exalt at most even for top-tier rolls. If you like the labyrinth and can do it fast, some serious money can be made this way.