Deign means to reluctantly agree to do something you consider beneath you. When threatened with the loss of her fortune, an heiress might deign to get a job, but she might look down her nose at the people she'd have to work with.
Deign has the same origins as dignity. Both descend from the Latin word, dignare, meaning "to deem worthy." If you deign to do something, you don't feel it's worthy of your lofty stature, but you do it anyway — it's like you're doing someone a really big favor. Instead of admitting his wrongdoing, the politician who is accused of taking bribes might indignantly declare, "I won't deign to dignify your ridiculous accusations with a response!"