The term "clone" is ambiguous (though the Java class library includes a Cloneable interface) and can refer to a deep copy or a shallow copy. Deep/shallow copies are not specifically tied to Java but are a general concept relating to making a copy of an object, and refers to how members of an object are also copied.
As an example, let's say you have a person class:
class Person {
String name;
List<String> emailAddresses
}
How do you clone objects of this class? If you are performing a shallow copy, you might copy name and put a reference to emailAddresses in the new object. But if you modified the contents of the emailAddresses list, you would be modifying the list in both copies (since that's how object references work).
A deep copy would mean that you recursively copy every member, so you would need to create a new List for the new Person, and then copy the contents from the old to the new object.
Although the above example is trivial, the differences between deep and shallow copies are significant and have a major impact on any application, especially if you are trying to devise a generic clone method in advance, without knowing how someone might use it later. There are times when you need deep or shallow semantics, or some hybrid where you deep copy some members but not others.