Closing a credit card can lower your credit score because your amount of available credit across cards decreases. But if you know which question to ask your credit card issuer, you might be able to avoid that.
Ask to have your credit limit transferred to another card with the same issuer. This can be especially helpful if you want to ditch a credit card that has an annual fee but has an impressively large credit limit. If you can get at least a portion of that limit transferred to another card, you can feel more confident closing that card and washing your hands of that annual fee. Your score may fluctuate slightly for a few months depending on the length of time you held the now-closed card, but your positive utilization should bolster your score for the long run.