Indelicate things are offensive or rude. If you tell an indelicate joke to your sweet, old-fashioned great-grandmother, it'll make her blush.
Many people would consider the question, "How much money do you make?" to be an indelicate one — it's tactless to ask it. Dirty jokes are indelicate because they're in bad taste, likely to offend someone. The adjective indelicate uses the "not" prefix in- along with delicate, which here means "tactful and considerate."