Prison v. Jail
First of all, failure to appear (in the U.S.) would led to incarceration in jail (a local government facility allowing for detention typically up to one year for misdemeanors), rather than prison (which is typically a state or federal government institution incarcerating people for a year or more pursuant to a felony conviction). A jail is typically run by the county sheriff or a city's chief of police. A prison is typically run by a warden appointed by a state department of corrections.
A civil court never sends anyone to prison for failure to appear, or for that matter, as a sanction for any purpose. Even a multi-year remedial civil contempt sanction is served in jail rather than prison.
When Can One By Jailed For Failure To Appear In A Civil Matter?
Second, failure to appear can only lead to incarceration if:
(1) the person who fails to appear in either subject to a subpoena requiring that person to appear, at a date and time certain in a particular place, as a witnesses, that is personally served upon them (or where personal service is waived by the person) (this is the same in civil and criminal practice), or
(2) the person is served with a judgment creditor's interrogatories to be answered in person (which are the moral equivalent of a subpoena) and must be served in the same manner, or
(3) the person fails to deliver a person pursuant to a writ of habeas corpus (which is the moral equivalent of a contempt of court finding and very uncommon) that has been served in the same manner, or
(4) the person has been personally served (or has personally waived such personal service) with an order to show cause (at a specific date, time and place) why they should not be held in contempt of court. The precise name of this document can vary and could also be called a "writ" or a "citation", rather than an "order to show cause". Citations for traffic offenses or other ordinance violations or civil offenses are functionally equivalent to (4).
Of course, in each of the cases (1) to (4) you obviously aren't incarcerated at the moment that you fail to appear as the whole problem is that you aren't there. Instead, in each of those cases, a warrant for your arrest is issued by the court and you aren't actually incarcerated until law enforcement manages to arrest you before the warrant is vacated because you do appear, or the warrant is vacated by stipulation of the parties approved by the court, or is vacated because the matter you were required to appear for is now moot (e.g. in the case of judgment creditor's interrogatories, the judgment is paid in full).
How Long Is One Incarcerated For Failure To Appear?
In cases (1)-(4) the duration of incarceration is only from the time of arrest until the person can be brought before the court at the soonest possible time to do whatever they were supposed to have shown up in person to do and failed to appear at, often a matter of a few hours and rarely more than a long weekend. These forms of incarceration are for the purpose of compelling your appearance in court by force, not for punishing someone for not showing up.
Other Reasons People Are Incarcerated In U.S. Civil Courts
One can also be incarcerated in a civil court because one is found to be in contempt of court after one has appeared and been found by the court in an in person hearing to be in contempt of court, but incarceration for contempt of court itself, as opposed to failure to appear at a contempt of court hearing, is never a sanction imposed for failure to appear.
Other Sanctions For Failing To Appear
Also, keep in mind that failing to appear can result in other consequences besides incarceration, the most common of which is to be found in default causing you to lose you legal case or whatever other legal matter was pending as a result of your failure to appear.
Another common sanction for failure to appear (e.g. at a deposition when you are not under subpoena) is a monetary sanction to compensate others for the attorneys' fees incurred due to your non-attendance, and a determination that facts that could have been established if you had appeared will be deemed to be true regardless of reality.
Civil Law Systems Compared
In "civil law" countries whose legal systems are not based upon the English common law, judges do not have a generalized contempt of court power and the scheme by which people are sanctioned for failing to appear in civil matters is different.