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A demand is a noble's request that your dwarves build a specific item of furniture in one of their rooms. Most nobles can make demands, and the number they make varies based on how important they are - see here for more details.
When a noble makes a demand, you will have about half a year to fulfill it. Demands are always for furniture, and can be for either furniture of a certain type, like a table, or furniture of a certain material, like copper items. Sometimes they specify both, for example "green glass window in dining room".
Demands are announced at the bottom of the screen, but if you miss the message, you can see if a noble is demanding anything on the noble's screen. If the uppercase bracketed word '[DEMAND]' next to a noble's name is gray, he is making no demands. If white, his current demands have been fulfilled or surpassed. If brown (or yellow when highlighted) the noble has stated their demand and has given you time to complete it. If red, the demanded is expected to have been completed already.
Fulfilling the demand gives the noble a happy thought. If a demand goes without being fulfilled, the noble will be "angered at the state of demands" and receive an unhappy thought, but no dwarves will be punished. Eventually, an announcement will be made that the noble has simply "forgotten" the demand.
Demands should not be confused with mandates.