Warning. A large part of the appeal of this game is the jokes. Spoilers below.
At one point when Stanley is going through the Half-Life 2 map, the Narrator gets so angry, so he starts making fun of Stanley, giving us one of the best lines on the mod:
Narrator: Stanley was fat and ugly, and really really stupid. He probably only got his job through some sort of family connection, that's how stupid he is. That, or with drug money. Also, Stanley is addicted to drugs and hookers.
In the HD Remake, this line is moved to the broom closet (see below).
If you stay in the boss' office for too long, the Narrator gets increasingly impatient and irritated, repeating the code insistently to make you get a move on.
"One. Nine. Five. Seven."
"Two. Eight. Four. Five."
Inversely, if you do the code before he says it, he'll get agitated and then suggests you calm down for a few moments and listen to new age music. On your next run through, he starts opening the secret door as soon as you enter the office, and then on the runs after that he starts opening it as you approach the office.
The newest trailer manages to find a very effective balance between the two, while also parodying Let's Players everywhere.
In the demo for the remake, you'll eventually come across a room that has nothing in it but a button with a number 8 on it, that causes a voice to say "eight" every time you press it. If you keep pressing the button after the narrator tells you to leave, he'll grow increasingly bewildered about how enthralled you are with the "game".
There are also uniquedemos given to Revision3 games and Steam Train, with additional narrator comments towards Adam Sessler and Danny and Ross respectively.
[when Adam tried to access the Final Choice the first time] No no no no no, Adam! What are you doing? You can't make that decision yet! We haven't played the demo! Is this what they call 'journalistic integrity'?
[when the Steam Train conductors "crash" the demo] Did the two of you do this? Goddammit Ross!
In the HD Remake, you can get into the broom closet, which annoys the narrator the longer you remain in there, until he concludes that you have died from some sort of malady and picks off where he leaves off "with a second player" when you leave it. Going back in has him just give up on everybody.
If you restart the game and enter the broom closet again, the Narrator gets very upset and doesn't even try. On the third time you pass the closet, the broom closet is boarded up.
The Confusion ending from start to finish. The game goes increasingly off track and the narrator becomes more bewildered as you progress.
The Stanley Parable Adventure Line™ from said Confusion ending. If the subtitles are turned on, every reference to The Line™ has a trademark symbol following it. The Line™ starts off seemingly normal, going across floors with Stanley following It™ while the Narrator starts playing adventure-like music in the background. But then It™ starts going crazy, going across walls and taking small detours, even through a corridor in between a set of pillars bigger than the room It™'s in at one point. Eventually, It™ betrays the Narrator by leading back into the Monitor Room, in which the Narrator stated that Stanley was not supposed to be there yet and considered it a spoiler.
And then after the next restart, when the Narrator takes Stanley into unexplored territory, The Line™ manages to burst through the ceiling and down through the corridor, but leaves in the same way after a while.
Made even funnier since the narrator tries to ignore It™ and tells The Line™ It™ had Its™ chance.
Another funny note about the extradimensional corridor mentioned above? It connects to a seemingly infinite storage locker set aside for nothing but peer reviews of Employee 432! (Thank goodness that isn't you!)
The Narrator's song, unlocked if you hop onto a co-worker's desk and out the window as soon as you leave the office.
The game's soundtrack on Bandcamp has a transcript of the lyrics of the song Following Stanley (the adventure music in the Confusion ending).