JOrtho (Java Orthography) is a Open Source spell-checker entirely written in Java. Its dictionaries are based on the free Wiktionary project and can therefore be updated by virtually any language. The library works with any JTextComponent from the Swing framework. This includes JTextPane, JEditorPane and JTextArea.
Java implementation of the statistical spelling checking algorithm by Peter Norvig which he described in his Essay "How to write a Spelling Corrector".
GNU Aspell is a Free and Open Source spell checker designed to eventually replace Ispell. It can either be used as a library or as an independent spell checker.