(Note: This page discusses the crimes and torts in general. Laws vary by location. These wordings are the terms in which the crime/tort is considered, though.)
Making a false accusation is actionable as "Malicious Prosecution," which is a tort, not a crime, for some reason. It cannot lead to arrest, merely to civil court where a complainant can try to get money for damages.
"Malicious Prosecution" (tort) is made against whoever "initiated" the original false charges. Usually this is police because they usually initiate charges. It is available for private prosecution, too, though.
To make this tort, the complainant has to show:
There was manifestly no reason for the initial charge (they knew the charges were false or they knew there was no reason to believe the charges were true)
There was malice / wrongful purpose in the decision to launch the initial false charge
(ie if a person is falsely charged but is found innocent, that is not enough to "prove" "malicious prosecution." You need more.)
Often, this charge is used when the initial false charge is made. This makes a court less likely to find guilt in the initial charge or could cause the initial complainant to drop their charges.
A weird thing is that apparently if the initial false charge succeeds you can't counter sue for making the false charge, it seems. You can't file the lawsuit in civil court until the initial charge is resolved in criminal court, it seems.
Also, "malicious prosecution" will likely fail against an official like the DA (county prosecutor in many cases) because you have to prove somehow they had malicious intent, and they are also often entitled to immunity even when its proved after a person has been convicted that they were actually innocent.
Another recourse is "Libel/Slander"
Many places, accusing someone of a crime falsely is considered "defamatory per se" or "actionable per se." That means that if someone accuses you falsely of a crime, the public authority considers that "harm is taken as a given" to you, and that your reputation is presumed to have been harmed.
How much liability a person has when they slander/libel another person varies by location. They might be liable for any damages that stemmed from their statements, including money lost as a result of problems/loss of work and inability to secure new work, harm to reputation, as well as less concrete damages such as embarrassment, mental anguish, humiliation.
Another recourse is "Blackmail" if the person made the false accusations after trying to blackmail someone else.
"Blackmail" is a crime and the charge leads to arrest. It's often a "Felony." In some places it is its own charge, in other places it is considered a form of "extortion or coercion."
In "Blackmail," the blackmailer has information (the information can be true or a lie, it makes no difference) that can damage the other party, and the blackmailer uses threats to cause the other party to do something they want them to do (something illegal or legal, it makes no difference). The central element of "Blackmail" is that the blackmailer's intent to get money or something else of value from the other party.
The threat made by the blackmailer to release information can include: to accuse someone falsely of a crime; to report someone's involvement in a crime; to reveal private or sensitive information about someone that will cause them embarrassment or financial harm.