getting a program like fireworks is the easiest way of producing gifs, and it comes out in better quality than a freebee.. You can also search on the internet for a free program called "Microsoft GIF animator"
All of the above is good advice, there are a host of programs that can make GIF animations, the one you use really depends on what you feel comfortable using. Just about all the major graphics programs have the capability to make them, or have companion applications that do (Example: Adobe Photoshop can not, however ImageReady performs that function. ImageReady is Adobe's web graphic companion to Photoshop and is very similar to Macromedia Fireworks)
But what is a GIF animation. Basically, it is a series of Graphic Interchange Format (GIF) images that have been stacked one on top of the other. Imagine one of those flip books you used to make as a kid, except now make it digital. You are limited to 8 bit color (256 colors) but since the creation of the GIF89a standard, GIF files support one color transparency, making them popular with early web pages.
As far as creation goes, it really depends on the program you are using. Some may support a robust set of drawing tools as well as some of the more standard animation tools, like tweens, while other "freebie" applications may only allow you to set up a GIF animation using already made images.