This role is the Mafia version of the Jailor. It has the ability to jail and role-block one player each night that follows a day with no lynch. The Kidnapper can speak anonymously with the jailed person at night, and optionally execute them.
If a Mayor reveals, it is probably in the Mafia's best interest to kidnap and execute him. Careful as this may strongly indicate that there's a Kidnapper in play
Kidnapping an Amnesiac can let you talk with them privately and anonymously and maybe convince them to take on a Mafia role.
Jailing either an Investigator or Sheriff can give the same effect as role-blocking them. By jailing them, you make them useless, slowing down the Town.
Kidnapping the following roles both wastes a night, as well as gives them unnecessary night immunity:
Mayor: Mayor can not normally do anything during the night, and you would only be giving him night immunity (Unless you plan to execute him)
Survivor: Persuading them to side with the Mafia at a later point will give you the advantage of winning if there is supposedly a tie between the Town and the Mafia. They might even be thankful for the night immunity, but it wastes a night for the Mafia.
Citizen: As Citizens do nothing during the night, this will give them free night immunity and waste your night.
Mason: The only benefit of kidnapping a Mason would be to prevent them from talking with their teammates, but this is generally seen as quite useless.
Jailing a fellow Mafia member (if the save permits) without charges or without a reason to use a charge may be wise if there is no need for the Kidnapper to jail someone that night. This ensures that they will survive the night (short of an invulnerability-piercing Arsonist attacking them) and continue to be an asset to the team.
Do not always execute instantly when an important role reveals to you, this may indicate there's a Kidnapper in play and other captives might not reveal useful information. Try to gather clues from your captives.
Listen to your teammates - they can help you decide whether to execute or not. Try not to answer them back at the night chat, as your captive might find it out of context and understand that you are actually a Kidnapper.