Excellent question. The answer depends on the substantive law of the state to which the facts are applied. I am licensed in Oregon, so let me answer the question under Oregon law. A jury can only award the types of damages which are pleaded in the case (usually the complaint). In Oregon, their are economic, non-economic, and punitive damages. Economic damages and non-economic damages are described by statute. They are both "compensatory." The person suing for those damages take the money by receiving a judgment from the court, even in jury cases.
Punitive damages are distinctly different in Oregon. For punitive damages, by statute in Oregon, a plaintiff (person bringing suit) must have permission from the court before punitive damages can even be alleged, always by motion. For punitive damages only, the state is entitled to a large portion of the punitive damages awarded. Recent Oregon case law has examined whether, and in what circumstances, a party receiving a punitive damage award can negotiate continue negotiating settlement after a jury's verdict, prior to the court's judgment. Often, after a big case, judgment is not submitted immediately on the jury's verdict. During the time from verdict (from the jury) to judgment (from the court), plaintiffs have successfully negotiated settlements whereby punitive damages are partially or entirely withdrawn, over the state's vigorous objection.
Note that the above answer may change, depending on the state law which is controlling in the substance of the lawsuit. That is often, but not always, the law of the state where the lawsuit is filed. For example, a person travels to a neighboring state and commits some civil act which ought to be punished under the laws of that neighboring state. If the person is sued in that other venue, where the act occurred, the law of that state should control. Similarly, if the act occurred in that neighboring state, but for whatever reason the person was sued in Oregon, arguably the substantive law of the neighboring state should control. That is a conflict of law question which would need to be researched, and may be among the considerations in determining whether to file in, or avoid, a particular state court.
In most civil suits, the plaintiff seeks compensatory damages to compensate them for their damages, and sometimes also punitive damages, based on the defendant's net wealth, to punish the defendant for their tortious and wilfull behavior. There are also "special" damages which are a subset of compensatory damages, and also legal fees and costs.
In certain cases, like civil contempt, there are fines that get paid to the court for disobediance of a court order, but in general, the plaintiff seeks money for themself.
See a lawyer to frame and plead the case proeprly - the inception of a case is the most important time.
Disclaimer: I'm only licensed in CA. Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.