Free Spell Check Source Code, Libraries, Delphi Components, ActiveX Controls

Enchant
Enchant is a wrapper around a number of spell checking libraries that enables you to use a standard API to access those spell checkers. It requires you to have the backend libraries. Supported backends include Aspell, PSpell, Ispell, MySpell, Hspell and AppleSpell (a number of which, if not all, are listed on this page). It is licensed under the GNU LGPL but with an additional note permitting any backend plugin to be loaded.
Hunspell
Hunspell is the default spell checker used in OpenOffice, a free office suite and other software. It can handle languages with complex word compounding and character encoding, such as the Hungarian language. It is licensed under the GNU GPL, the GNU LGPL and the MPL.
Pspell
PSpell, the Portable Spell Checker Interface Library, "provides a generic interface to spell checker libraries installed on the system" (quote from the Sourceforge project page). It is licensed under the GNU Library General Public License (LGPL).
GNU Aspell
GNU Aspell is an open source spell checker that can be used either as a library which you can integrate in your programs or as a standalone speller. It is able to suggest misspelled words (supposedly better than any other spell checker around, or so their site claims), check documents in UTF-8, supports multiple dictionaries simultaneously, handles personal dictionaries, etc. The code is in C++, but you can access the library via its various language bindings as well, so that it is possible to use from Perl, PHP and Ruby. Dictionaries are available for a wide variety of languages, too many to list here. It runs on POSIX systems (Unix, Linux, FreeBSD, OpenBSD, etc). If you are looking for a Windows port, see elsewhere on this page.
GNU Aspell (Win32 version)
GNU Aspell (Win32 version) is a port of the original GNU Aspell to Windows (doh!). Please see the description for Aspell elsewhere on this page for more details about Aspell in general. There are linkable libraries here that you can use with MinGW gcc (see the Free C/C++ Compilers page) and Microsoft Visual C/C++, a Delphi unit (see the Free Delphi Compilers, Free Pascal Compilers page), etc. Precompiled dictionaries for Breton, Catalan, Czech, Welsh, Danish, German, Greek, English, Esperanto, Spanish, Faroese, French, Italian, Dutch, Norwegian, Polish, Portuguese, Romanian, Rusian, Slovak, Swedish and Ukranian are also available from the site. (If your language isn't listed here, you might want to check the main Aspell page for more dictionaries.)
International Ispell
Ispell is not strictly a library, but a standalone spell checker. However, since it comes with source code, you might want to take a look at it to get ideas for implementing your own spell checker (if that's your cup of tea). It comes with British and American English dictionaries, but there are dictionaries for numerous other languages available as well. It works on Unix type systems (including Linux), with ports for MSDOS, Windows and OS/2.
Jazzy: The Java Open Source Spell Checker
If you write Java programs, you can use Jazzy to add spell checking facilities to them. At this time, it comes with an English dictionary.
A Free Spell Checker with a Dictionary Editor Program
This spell checker is designed to integrate with your MFC application (in other words, it's a Windows spell checker). It comes with a UK English dictionary as well as a dictionary editor.
LS Speller
LS Speller contains spell checker components for Delphi 5, 6 and 7 that allows you to add spell check facilities to your program. The spell checker works with ispell dictionaries as well as dictionaries that come with Microsoft Office 95 or 97.