To do something respectably is to act in a proper, presentable, or acceptable way. If your dad asks you to dress respectably for a family wedding, he probably wants you to wear formal, tidy clothes.
Sometimes respectably means "decently," or "politely," as when a teacher says, "Stop tap dancing on my desk! We behave respectably in this classroom." Other times, it means something more like "not bad," as when a business owner says, "Our sales team did respectably this quarter." At the root of this adverb is the Latin respectus, "regard," or "act of looking back at one."