The Emotions' Headquarters is the central location of Riley's mind. It's the room where the emotions stand behind a large control panel and each button makes Riley do a certain action (when Anger pulls two levers at the same time, Riley has a violent outburst). Its appearance is that of a condo. The walls are filled with different memories that Riley has, and there are even couches and a staircase. There's apparently a long vacuum tube that can suck out anything. When Joy and Sadness ended up getting sucked out of Headquarters, Anger, Disgust and Fear were left to control Riley in their absence.
"Headquarters" is a pun.
The console only consisted of a single button when Riley was born.
When Riley gets a Brain Freeze, the whole headquarters and anything inside it will freeze, including the emotions.
At the end of the movie, the control panel became much larger and it has more multi-colored buttons and levers and they make up more than just one expression Riley can show up. There's also a big red alarm for puberty.
All the emotions live and work inside the headquarters. It is implied each of the emotions has their own sleeping quarters, but they are not seen in the movie.
All five of the emotions have lockers located under the stairwell. Since the stairwell curves, each locker has an escalating height which matches the emotion of the same height. Anger has the smallest locker, since he is the shortest, and Joy has the biggest one, since she is the tallest of the five emotions.
Each emotion's locker has a symbol with the color that represents which emotion owns which locker.
In a possible second movie, the color is red all over headquarters.
When the console starts to turn black, all the controls become inoperable. Likely, this signifies the feeling of emotionless, or the console shutting down.
3D render wide view
Headquarters (on left) is apparently one VERY tall building