If you're on a Mac, SimpleText is probably sitting on your hard drive. It's a free text editor and that's all you need to write HTML.
If you're using a PC, there's a good chance you have WordPad pre-installed. Like SimpleText, WordPad is a simple and free text editor.
BBEdit Lite is a text editor for Macintosh that has features specifically for making Web pages. BBEdit Lite is the free version of BBEdit.
EditPlus is an HTML-enhanced text editor like BBEdit Lite, but it's for Windows. It costs US$20.
GifBuilder is a very popular freeware GIF animation tool for the Mac.
GIF Construction Set is shareware (it costs US$20) from Alchemy Mindworks and an excellent GIF animation tool for Windows.
Adobe Photoshop is currently the de facto industry standard software program for processing images.
Macromedia's Dreamweaver is a heavy duty tool for making sophisticated Web sites.
To move files from computers to servers, there are two easy FTP programs to use. For the Mac, Fetch is free for educators and non-profits (US$25 for everybody else); and CuteFTP is the Windows FTP solution.
Webmonkey is Wired Digital's how-to guide for Web developers.
O'Reilly publishes a comprehensive series of reference books about the Internet and computing.