Each person gets a key. Thaat key is a specific level1 to a specific dungeon. So, in a group of 5 people, you have five keys2.
You decide which key you're going to do, zone into that dungeon, and stick the appropriate key in the slot (there's a little pedastal at the beginning of the Mythic dungeon). If you complete the key within the specified time, it increases by 1 level, and switches to a random dungeon3. If you complete the dungeon but not in time, it downgrades by 1 and switches to another dungeon. If you don't complete the dungeon at all, it downgrades by 1 but stays on the same dungeon. Since only one key can upgrade at once, if you're in a regular group, you'll probably pick whichever is the best key at the start of the week and just do whatever dungeons it turns into along the way (see also 3)
For higher levels, you definitely need to know what the boss mechanics are, and the main ability of more problematic trash mobs (often mechanics that you could flat out ignore at "base" mythic level). If you're tanking, you may also need to figure out specific routes, which trash packs to pull or skip etc, because each dungeon also has a "Trash percentage" quotient that must be completed. Some trash mobs are worth much more than others, and some are notoriously "not worth it" due to their difficulty4.
The m+ affixes can significantly change how certain mechanics interact, and I've not figured out all the permutations yet for this season, but some of them are... awkward, and hopefully sufficiently unintended to be fixed (e.g. crystals spawning on packs that will then instantly absorb several or all of them when later pulled)
1 The highest of (the highest level you completed in time last week) or (1 down from the highest level you completed by not in time), with an additional penalty of 1 per week if you didn't complete any at all.
2 So, in theory, you get up to 5 choices, but every week so far this season, our choices have been either Siege of Boralus or Necrotic Wake. RNG sucks sometimes.
3 And in our group, routinely, it switches to the other one of (Siege, Necrotic) that you didn't just do. We have yet to see a single instance of either of the Azh Kahet dungeon keys.
4 On weeks with the Fortified affix, there were trash packs we saved Bloodlust for, rather than bosses. Some of them are boss-level difficulty.