The Mysterious Door is an area found in Snowdin Forest, south of the right side of the room with the icy X's and O's puzzle. The door, which bears the Delta Rune, is surrounded by glowing mushrooms and floating particles. Glyde can be encountered in this room before fighting Papyrus.
Beyond the door lies the "developer room" which includes the Annoying Dog who represents Toby Fox, a computer open to bad posts that the dog has made, a jigsaw puzzle, the rope that was probably used in Papyrus's Gauntlet of Deadly Terror, a "text-to-speech" converter telling the player that the dog somehow programmed the game, and a dog-shaped husk of Dog Residue.
The following dropdown may contain spoilers.
[When interacting with the game case]
(By barking with text-to-speech on, the dog accidentally programmed a whole game)
(Hmmm... seems fine.)
[When interacting with the rope]
(Looks like this dog does all its own stunts.)
[When interacting with the puzzle]
(Finally, the pieces of the puzzle are coming together.)
(The jigsaw puzzle.)
[When interacting with the dog to the left]
(You tried to pet the dog, but it collapsed into a pile of fluff...)
(Seems like it was actually just a dog-shaped husk of dog residue.)
(It's a pile of dog residue.)[When interacting again]
[When interacting with the computer]
(Looks like this dog makes a lot of bad posts on-line.)
(Text to speech is on.)
(Seems like the dog barks, and it translates to these bad posts)
(It doesn't really know what it's saying.)
(Seems like most bad posts on-line, are actually this dog.)
(Thinking about this...)
(Brings you relief.)
[When interacting with the dog in the middle]
(It's a small white dog.)
(It's fast asleep...)
(Fight the dog?)
[If "No" is selected]
(Let sleeping dogs lie, instead of fighting them.)
(That's how the saying goes.)
[If "Yes" is selected]
(Can't fight the dog.)
(Seems like the fabric it's sleeping on has too many holes in it.)
(Seems like the dog needs to "patch" the fabric.)
(Then you can fight the dog.)
(Upon closer examination, the holes in the fabric seem to be growing.)
(Might take a while for the dog to fix all of them.)
(You used the key, the door's lock clicks..... as you fail to fit the key into it.)
The dog made up of sludge that then falls apart may be a reference to page 2832 of the webcomic, "Homestuck."
The piece of fabric that the Annoying Dog is lying on states that "the dog needs to 'patch' the fabric." This is most likely referring to the actual game itself, as Toby Fox needs to patch the game up.
If the mystery key is used when facing the door, a click is heard, but the door does not open.
In the room plays a music called mus_dogroom.ogg. This song, however, isn't in the Undertale Soundtrack. It can be heard here.
↑You used the Mystery Key. The door's lock clicks...... as you fail to fit the key into it.