Demands are requests for various things Nobles want that you have half a year to fulfill. By comparison Mandates are requirements you must follow, both have similar repercussions.
If you fulfill a Demand or Mandate the Noble in question will receive a Happy Thought. Failing to do so will result in an Unhappy Thought. More over and unfulfilled Mandate may get sentenced to punishment for the 'violation of production order' crime.
Dwarven productivity is generally determined by the amount of Happy/Unhappy Thought they have. Since most nobles are not productive this isn't usually an issue. However, Dwarves with a large number of Unhappy thoughts can enter into (SPOILERS) Fell or Macabre moods (the former resulting in the death of another dwarf). Worse, they can also lead to Tantrums. The issue with Tantrums and Fell moods is that the death of a dwarf (any dwarf) can easily kick off a Tantrum Spiral.
So do us all a favor, and meet your Noble's needs.