What did bounty hunters do?

Answer 1

when do bounty hunters get involved in catching criminals

Modern day bounty hunters or more appropriately called, bail fugitive recovery agents, only work when hired by a bondsman to locate and return the defendant who has missed court back to the courts jurisdiction.

Unlike Duane "Dog" Chapman, bounty hunters can not go to foreign countries or "arrest" anyone they have not been hired to arrest. Bondsman most get authorization to a properly certified or licensed bounty hunter to locate and arrest a bail skip.

Bounty hunters have no authority to arrest anyone other than the bail skip they have been hired to locate and arrest.

people who hunt criminals who are "wanted"...they get money once the criminal is captured

In the past there were bounties placed on certain animals. People would be paid a certain price for every scalp or animal skin they brought in. Wolves were almost hunted to extinction through the use of bounties. Unfortunately, the concept was also applied to American Indians during the French and Indian War.