Officer Jenny is not a single entity. Rather, there are many Jennys working as police officers across the various regions of the Pokémon world.
Generally, Officer Jennies are brave and determined, but mostly ineffective in their police duties. They fight against the likes of Team Rocket, Team Magma, Team Aqua and Team Galactic - among other miscellaneous criminals. The town/region where a certain Jenny comes from can be discerned by the symbol on their hats which always differs depending on location.
Although Jennies appear mostly identical, there is some minor divergence among them, as evidenced by Brock, who is able to recognize different Jennies on numerous occasions. Brock appears to be the only character who can tell them apart. However, each Jenny has a different symbol on her hat that indicates location. There seem to be two Viridian Jennies, as during the Indigo League, two Jennies with the same hat symbol were seen side-by-side. (We know that the symbol in question—a star—indicates Viridian due to previous episodes.) One of the Viridian Jennies maintains a fierce rivalry with the Jenny of Eterna City.
Although there are many exceptions, Officer Jenny will generally have Growlithe to use as a police dog.
There is a very distinct parallel between the families of Officer Jenny and Nurse Joy. This may indicate some sort of relation or deeper connection, and fans have sometimes speculated about such. However, the only thing known about the two families is that they are apparently on good terms.
Ash’s Squirtle was given to an Officer Jenny who fights fires. It is the leader of a group of four other Squirtle known as the “Squirtle Squad,” who serve as the local firefighting team.
A young Officer Jenny named Marble appeared in Luxray Vision!. She was in charge of looking for “The Bandit Beauties Three.”
An Officer Jenny from Kanto appeared in Stealing the Conversation! with a Chatot as her partner. Apparently, she had come all the way to Sinnoh to capture Jessie, James, and Meowth.