The title of this article is conjectural as this subject has no known canonical name in the DCEU.
The Joker's gang is a large group of career criminals operating in Gotham City. They are lead by the murderous Joker (though Harley Quinn has been known to lead them whenever he is busy or incapacitated). The gang commit acts of extreme crime that almost always lead to large amounts of death and destruction, usually for the Joker's own amusement.
Many of them are small-time robbers, hitmen, shanghaied gangsters, mentally-ill escapees of Arkham Asylum and dangerous criminals. Most, if not all, of them suffer from some form of mental illness; The kind of minds that the Joker attracts.
Although they are usually referred to as a gang, some people have noted that they are similar to a military army and others have described them as a cult.