"Too many objects" is the problem that FlyWeight attempts to address.
In java.lang.String example, note that the individual characters that make up the
string are not modeled as objects because doing so would mean creating a large number of objects affecting performance, footprint and more than anything, manageability. String object instead represents an aggregate of character instances, acts as a single object instance while still allowing sharing of those characters at the finegrained level.
Note that "sharing" is also an important criteria here, not just manageability.
Flyweights are used in scenarios where object identity is determined at a conceptual level ie., two strings are equal if their character sequence is equal and this makes sharing possible. This refers to the concept of "FlyWeight pools" that enable reuse of constituent entities. Every String has an extrinsic state which is its presentation as a sequence of characters, as a
single object whereas its intrinsic state may be more distinct ie., the individual bytes that make up the string.
Here are a few other examples of the FlyWeight patter
Infinite number of points on an arbitrarily shaped curve. Seats in an aircraft( hint..hint.. 
) An index table of entries( in a book, for example ) I strongly recommend reading the GOF Design Patterns book if you can get hold of a copy. I found the explanation of FlyWeight pattern very easy to understand, like every other pattern in the book.
HTH
