What is a bench warrant? Is there a difference between a bench warrant and a regular arrest warrant?
Here's a mnemonic device to start you off: A "bench warrant" usually means someone didn't have their backside on the bench when they should have. That's the bench in front of the judge, not the park bench where pigeons can be fed.
Speaking more technically, a bench warrant is issued by a judge when a defendant violates the rules of the court. Most often, the defendant has simply failed to show up. Once a bench warrant is issued, however, the police can treat it like any other arrest warrant—and use it to bring the defendant back in front of the judge. (For more on failing to appear in court, see Bail Jumping—or Failing to Appear After Bailing Out.)
By contrast, a police officer initiates the arrest warrant process, by filing a statement with the judge that explains why the officer believes that the person named has committed a crime (in legalese, the officer is showing "probable cause" to arrest the person). If convinced, the judge signs the warrant, and the police can make the arrest. This process doesn't police officers to run to court for arrest warrants for every purse-snatcher they see. Many jurisdictions allow officers to call the on-duty judge to request a warrant, and many arrests do not require a warrant at all.
The most typical use of an arrest warrant is when an officer wants to arrest someone in their home. Simply busting down the suspect's front door in such a non-emergency situation would be neither polite nor legal.