Constructors in Java can be categorized as default or parameterized constructors. Default constructors require no parameters, while parameterized constructors allow passing parameters to initialize object attributes. Constructors are used to initialize objects and are called automatically whenever a new object is created. Constructor overloading allows defining multiple constructors that differ in their parameters. There is no copy constructor in Java, but objects can be copied by assigning values in one object to another object or by passing one object to another object's constructor.