The Java tutorial says:

Terminology: Nested classes are divided into two categories: static and non-static. Nested classes that are declared static are simply called static nested classes. Non-static nested classes are called inner classes.

In common parlance, the terms "nested" and "inner" are used interchangeably by most programmers, but I'll use the correct term "nested class" which covers both inner and static.

Classes can be nested ad infinitum, e.g. class A can contain class B which contains class C which contains class D, etc. However, more than one level of class nesting is rare, as it is generally bad design.

There are three reasons you might create a nested class:

• organization: sometimes it seems most sensible to sort a class into the namespace of another class, especially when it won't be used in any other context
• access: nested classes have special access to the variables/fields of their containing classes (precisely which variables/fields depends on the kind of nested class, whether inner or static).
• convenience: having to create a new file for every new type is bothersome, again, especially when the type will only be used in one context

There are four kinds of nested class in Java. In brief, they are:

• static class: declared as a static member of another class
• inner class: declared as an instance member of another class
• local inner class: declared inside an instance method of another class
• anonymous inner class: like a local inner class, but written as an expression which returns a one-off object

Let me elaborate in more details.

Static Classes

Static classes are the easiest kind to understand because they have nothing to do with instances of the containing class.

A static class is a class declared as a static member of another class. Just like other static members, such a class is really just a hanger on that uses the containing class as its namespace, e.g. the class Goat declared as a static member of class Rhino in the package pizza is known by the name pizza.Rhino.Goat.

package pizza;

public class Rhino {

    ...

    public static class Goat {
        ...
    }
}

Frankly, static classes are a pretty worthless feature because classes are already divided into namespaces by packages. The only real conceivable reason to create a static class is that such a class has access to its containing class's private static members, but I find this to be a pretty lame justification for the static class feature to exist.

Inner Classes

An inner class is a class declared as a non-static member of another class:

package pizza;

public class Rhino {

    public class Goat {
        ...
    }

    private void jerry() {
        Goat g = new Goat();
    }
}

Like with a static class, the inner class is known as qualified by its containing class name, pizza.Rhino.Goat, but inside the containing class, it can be known by its simple name. However, every instance of an inner class is tied to a particular instance of its containing class: above, the Goat created in jerry, is implicitly tied to the Rhino instance this in jerry. Otherwise, we make the associated Rhino instance explicit when we instantiate Goat:

Rhino rhino = new Rhino();
Rhino.Goat goat = rhino.new Goat();

(Notice you refer to the inner type as just Goat in the weird new syntax: Java infers the containing type from the rhino part. And, yes new rhino.Goat() would have made more sense to me too.)

So what does this gain us? Well, the inner class instance has access to the instance members of the containing class instance. These enclosing instance members are referred to inside the inner class via just their simple names, not via this (this in the inner class refers to the inner class instance, not the associated containing class instance):

public class Rhino {

    private String barry;

    public class Goat {
        public void colin() {
            System.out.println(barry);
        }
    }
}

In the inner class, you can refer to this of the containing class as Rhino.this, and you can use this to refer to its members, e.g. Rhino.this.barry.

Local Inner Classes

A local inner class is a class declared in the body of a method. Such a class is only known within its containing method, so it can only be instantiated and have its members accessed within its containing method. The gain is that a local inner class instance is tied to and can access the final local variables of its containing method. When the instance uses a final local of its containing method, the variable retains the value it held at the time of the instance's creation, even if the variable has gone out of scope (this is effectively Java's crude, limited version of closures).

Because a local inner class is neither the member of a class or package, it is not declared with an access level. (Be clear, however, that its own members have access levels like in a normal class.)

If a local inner class is declared in an instance method, an instantiation of the inner class is tied to the instance held by the containing method's this at the time of the instance's creation, and so the containing class's instance members are accessible like in an instance inner class. A local inner class is instantiated simply via its name, e.g. local inner class Cat is instantiated as new Cat(), not new this.Cat() as you might expect.

Anonymous Inner Classes

An anonymous inner class is a syntactically convenient way of writing a local inner class. Most commonly, a local inner class is instantiated at most just once each time its containing method is run. It would be nice, then, if we could combine the local inner class definition and its single instantiation into one convenient syntax form, and it would also be nice if we didn't have to think up a name for the class (the fewer unhelpful names your code contains, the better). An anonymous inner class allows both these things:

new *ParentClassName*(*constructorArgs*) {*members*}

This is an expression returning a new instance of an unnamed class which extends ParentClassName. You cannot supply your own constructor; rather, one is implicitly supplied which simply calls the super constructor, so the arguments supplied must fit the super constructor. (If the parent contains multiple constructors, the “simplest” one is called, “simplest” as determined by a rather complex set of rules not worth bothering to learn in detail--just pay attention to what NetBeans or Eclipse tell you.)

Alternatively, you can specify an interface to implement:

new *InterfaceName*() {*members*}

Such a declaration creates a new instance of an unnamed class which extends Object and implements InterfaceName. Again, you cannot supply your own constructor; in this case, Java implicitly supplies a no-arg, do-nothing constructor (so there will never be constructor arguments in this case).

Even though you can't give an anonymous inner class a constructor, you can still do any setup you want using an initializer block (a {} block placed outside any method).

Be clear that an anonymous inner class is simply a less flexible way of creating a local inner class with one instance. If you want a local inner class which implements multiple interfaces or which implements interfaces while extending some class other than Object or which specifies its own constructor, you're stuck creating a regular named local inner class.

You are correct in assuming that the attribute access available to non-static inner classes leads to high coupling, hence to lower code quality, and (non-anonymous and non-local) inner classes should generally be static.

Design implications of decision to make inner class non-static are laid out in Java Puzzlers, Puzzle 90 (bold font in below quote is mine):

Whenever you write a member class, ask yourself, Does this class really need an enclosing instance? If the answer is no, make it static. Inner classes are sometimes useful, but they can easily introduce complications that make a program difficult to understand. They have complex interactions with generics (Puzzle 89), reflection (Puzzle 80), and inheritance (this puzzle). If you declare Inner1 to be static, the problem goes away. If you also declare Inner2 to be static, you can actually understand what the program does: a nice bonus indeed.

In summary, it is rarely appropriate for one class to be both an inner class and a subclass of another. More generally, it is rarely appropriate to extend an inner class; if you must, think long and hard about the enclosing instance. Also, prefer static nested classes to non-static. Most member classes can and should be declared static.

^{If you're interested, a more detailed analysis of Puzzle 90 is provided in this answer at Stack Overflow.}

It is worth noting that above is essentially an extended version of the guidance given in Java Classes and Objects tutorial:

Use a non-static nested class (or inner class) if you require access to an enclosing instance's non-public fields and methods. Use a static nested class if you don't require this access.

So, the answer to the question you asked in other words per tutorial is, the only convincing reason to use non-static is when access to an enclosing instance's non-public fields and methods is required.

Tutorial wording is somewhat broad (this may be the reason why Java Puzzlers make an attempt to strengthen it and narrow it down). In particular, directly accessing enclosing instance fields has never been really required in my experience - in the sense that alternative ways like passing these as constructor / method parameters always turned easier to debug and maintain.

Overall, my (quite painful) encounters with debugging inner classes directly accessing fields of enclosing instance made strong impression that this practice resembles use of global state, along with known evils associated with it.

Of course, Java makes it so that damage of such a "quasi global" is contained within enclosing class, but when I had to debug particular inner class, it felt like such a band aid didn't help to reduce pain: I still had to keep in mind "foreign" semantic and details instead of fully focusing on analysis of a particular troublesome object.

For the sake of completeness, there may be cases where above reasoning doesn't apply. For example, per my reading of map.keySet javadocs, this feature suggests tight coupling and as a result, invalidates arguments against non-static classes:

Returns a Set view of the keys contained in this map. The set is backed by the map, so changes to the map are reflected in the set, and vice-versa...

Not that above would somehow make involved code easier to maintain, test and debug mind you, but it at least could allow one to argue that complication matches / is justified by intended functionality.