When you do something nicely, you do it in a pleasant or charming way. Asking your brother nicely to borrow ten dollars is likely to work a lot better than demanding a loan.
A nicely dressed applicant has an advantage at a job interview over someone wearing ripped jeans. You can use the adverb nicely to mean "attractively," "politely," or "in a satisfactory way." It comes from nice, which originally meant "foolish," and whose meaning has changed in an unusually dramatic way over the years, from "stupid," to "fussy," to "delicate," and finally to "agreeable."