Hot pursuit is when law enforcement actively chases an alleged criminal (the suspect). The act of the suspect fleeing the scene and the law enforcement officer following them is considered to be a pursuit that is ongoing or fresh.
If a suspect leaves the scene of a crime, especially when they know that they are being followed or asked to stop by a law enforcement officer, it is a compounded crime. (The act of fleeing will be added to the other charges.)
Sometimes, police officers either witness a crime take place or respond to a call that a crime has taken place. In these circumstances, a chase of a suspected criminal can ensue. Such chases, known as hot pursuits, can involve running through neighborhoods, car chases, etc.
If a police officer or other law enforcement agent is in hot pursuit, they do not need to obtain a warrant or authorization to leave their jurisdiction. Such standard legal procedures are trumped by the need to apprehend the criminal and prevent them from escaping, to prevent the destruction of evidence or property, and to prevent harm to the public. For example, police officers are not going to try to get a warrant if they witness a bank robbery. Instead, they will pursue the suspects in a hot pursuit.