Note: Adobe discontinued Flash Player in December 2020.
One of the most popular ways to create animation effects on a web site is to use Flash. The official method for creating a Flash movie, ie a .SWF file, is to use Adobe's software. But it's difficult to use, and very expensive.
Giotto, from Vectorian, is a free Windows program for creating .SWF files. You can get it from http://vectorian.com/downloads/, where you'll also find a Quick Start guide and a collection of tutorials to help you get started with the program. There are also loads of pre-built effects and ActionScript snippets, so you don't have to write any code.
Giotto is an 8 MB download, and runs on all versions of Windows from 2000 onwards.