This is useful if you want the currently executing method to still throw the exception while allowing resources to be cleaned up appropriately. Below is a concrete example of handling the exception from a calling method.

public void yourOtherMethod() {
    try {
        yourMethod();
    } catch (YourException ex) {
        // handle exception
    }
}    

public void yourMethod() throws YourException {
    try {
        db.store(mydata);
    } finally {
        db.cleanup();
    }
}

It depends on whether you can deal with the exceptions that can be raised at this point or not.

If you can handle the exceptions locally you should, and it is better to handle the error as close to where it is raised as possible.

If you can't handle them locally then just having a try / finally block is perfectly reasonable - assuming there's some code you need to execute regardless of whether the method succeeded or not. For example (from Neil's comment), opening a stream and then passing that stream to an inner method to be loaded is an excellent example of when you'd need try { } finally { }, using the finally clause to ensure that the stream is closed regardless of the success or failure of the read.

However, you will still need an exception handler somewhere in your code - unless you want your application to crash completely of course. It depends on the architecture of your application exactly where that handler is.

You would use it to ensure some actions occur after the try content or on an exception, but when you don't wish to consume that exception.

Just to be clear, this doesn't hide exceptions. The finally block is run before the exception is propagated up the call stack.

You would also inadvertently use it when you use the using keyword, because this compiles into a try-finally (not an exact conversion, but for argument's sake it is close enough).

try
{
    TrySomeCodeThatMightException();
}
finally
{
    CleanupEvenOnFailure();
}

Code running in finally is not guaranteed to run, however the case where it isn't guaranteed is fairly edge - I can't even remember it. All I remember is, if you are in that case, chances are very good that not running the finally isn't your biggest problem :-) so basically don't sweat it.

Update from Tobias: finally will not run if the process is killed.

Update from Paddy: Conditions when finally does not execute in a .net try..finally block

The most prevalent example you may see is disposing of a database connection or external resource even if the code fails:

using (var conn = new SqlConnection("")) // Ignore the fact we likely use ORM ;-)
{
    // Do stuff.
}

Compiles into something like:

SqlConnection conn;

try
{
    conn = new SqlConnection("");
    // Do stuff.
}
finally
{
    if (conn != null)
        conn.Dispose();
}

In addition to what others have said, it's also possible for an exception to be thrown inside the catch clause. Consider this:

try { 
    throw new SomeException();
} catch {
    DoSomethingWhichUnexpectedlyThrows();
}
Cleanup();

In this example, the Cleanup() function never runs, because an exception gets thrown in the catch clause and the next highest up catch in the call stack will catch that. Using a finally block removes this risk, and makes the code cleaner to boot.

finally should have atleast a try block, catch is optional. The point of finally blocks is to make sure stuff gets cleaned up whether an exception is thrown or not. As per the JLS

A finally clause ensures that the finally block is executed after the try block and any catch block that might be executed, no matter how control leaves the try block or catch block.

Hence a finally should always be preceded by a try block.