Lock and Key is a sub-quest of Hall of Echoes Bound, part of the game's main quest. You must find the Hall of Echoes Sigil that will open the door to The Hall of Echoes by defeating Ba'al in the Broken Valley Mine. The mine is located near where the Quarry Waypoint Shrine was on the Broken Valley map prior to the region being poisoned. You may remember seeing people fleeing the site and barricading the main entry earlier in the game. In order to get in you'll need to find and unlock an alternate entry.
To gain access to the mine, you must stop at the Broken Valley Mine Tower first. Inside the building, you come face-to-face with two level 27 Black Ring members. Once you defeat them, go up all the flights of stairs (there several chests along the way, one of them requires a key which is found on a desk on the top floor). Here you find Williams, who tells you about an alternate way to enter the mine and gives you the key. At the end of the conversation a mini-quest, A Guild Without a Master, automatically begins. To complete it, simply escort Williams and the soldiers to the ground floor, protecting them from the invading 27 Black Ring members along the way.
Once outside, get back into dragon form and head southeast towards the mine entrance, which is marked on your map. The area is protected by towers, so be cautious.
After entering the mine, you can either loot it thoroughly while completing the quest (see Broken Valley Mine for details), or ignore much of the mine and focus on getting the Hall of Echoes Sigil. To get your objective accomplished as quickly as possible, follow the path from the entry until you get to a four-way junction. Note the locked door to your left (north), and continue straight ahead (east). The path soon curves north - you may want to quicksave here. Go north to a heavily populated area with level 28 Black Ring members, demons, and summoned fire skeletons. An officer named Amon will attack you here as well. Once he's dead, loot his key, then turn around and return to the locked door at the junction. Unlock the door and head north to another door (labelled "Maxos Door"). Save you game here - once you enter it will automatically and permanently close behind you. Inside you'll face Ba'al (level 30), who spawns a lot of demon helpers (level 28), some of which are summoners and therefore summon even more.
Defeat Ba'al and loot the Hall of Echoes Sigil to end the quest. This also grants "The Sigil, the Seal" achievement. You can exit the mine through a teleporter to the north.