Java

Clone (κλών) is a Greek word meaning “branch”, referring to the process whereby a new plant can be created from a twig. In biology it is about copying the DNAs. In real world, if you clone Marilyn Monroe, will you get a copy of her with same beauty and characteristics? No, you will not get! This exactly applies to java also. See how java guys are good at naming technologies.
In java, though clone is ‘intended’ to produce a copy of the same object it is not guaranteed. Clone comes with lots of its and buts. So my first advice is to not depend on clones. If you want to provide a handle / method to deliver a copy of the current instance write a kind of factory method and provide it with a good documentation. When you are in a situation to use a third party component and produce copies of it using the clone method, then investigate that implementation carefully and get to know what is underlying. Because when you ask for a rabbit, it may give monkeys!

Shallow Copy

Generally clone method of an object, creates a new instance of the same class and copies all the fields to the new instance and returns it. This is nothing but shallow copy. Object class provides a clone method and provides support for the shallow copy. It returns ‘Object’ as type and you need to explicitly cast back to your original object.
Since the Object class has the clone method (protected) you cannot use it in all your classes. The class which you want to be cloned should implement clone method and overwrite it. It should provide its own meaning for copy or to the least it should invoke the super.clone(). Also you have to implement Cloneable marker interface or else you will get CloneNotSupportedException. When you invoke the super.clone() then you are dependent on the Object class’s implementation and what you get is a shallow copy.

Deep Copy

When you need a deep copy then you need to implement it yourself. When the copied object contains some other object its references are copied recursively in deep copy. When you implement deep copy be careful as you might fall for cyclic dependencies. If you don’t want to implement deep copy yourselves then you can go for serialization. It does implements deep copy implicitly and gracefully handling cyclic dependencies.