Answer is a word of noble pedigree—it dates back nearly a thousand years in its original sense of “swear against” (from Old English andswaru). However, it’s bland and neutral, and a variety of synonyms with more precise connotations exist.
Reply often has a sense of a thorough reaction to a communication, though it can simply refer to an answer in general. Response, on the other hand, has a sense of “a prompt or spontaneous reply,” though it can also be applied to more extensive communication (sometimes clarified with a modifier in such phrases as “measured response”). Replication, a formal extension of reply, is usually employed only in legal contexts and is better known as a synonym for duplication.
Rejoinder and retort connote some tension in the communication: Rejoinder implies that the original statement was a criticism or an objection, while retort suggests that the reply is (or is perceived as) an attack; to point out the animosity involved, a retort is often described as “cutting” or “short.” Return can be used as a synonym for retort.
The colloquial term comeback, meanwhile, describes a quick response that attacks the person making the original statement and is meant to diminish or insult him or her. A take, on the other hand, is a subjective but emotionally neutral response to something—generally, a statement of analysis or opinion on an event or an issue. Though colloquial, take is commonly used in writing, followed by the preposition on (as in “What’s your take on the matter?” or “Smith offered his take on the incident.”).
A rebuttal is not a response per se, but it is an act of disproving by offering an argument or evidence against a statement. Similarly, a refutation is a claim that something said or written is false (the distinction between rebuttal and refutation is that the latter is not necessarily supported by evidence or an effective argument.)
An informal abbreviation expressing a request for an answer is RSVP, from the French phrase “Répondez s’il vous plait.” Though plait means “please,” many who use the phrase are unaware of the literal translation and redundantly ask respondents to “Please RSVP”; pointing out this duplication seems to be a lost cause and will probably result in a rejoinder or a retort.