Graphics editors come in many flavors and prices. The important thing is that they need to be able to handle JPG and GIF formatted graphics. These are the two kinds of graphics you can use on the web. Any graphic that you download from the web will be in one of these formats. So if you create any yourself, you need to make sure you save them in one of these formats. True enough, another format - PNG is on the horizon, so look for that in the newer browser capabilities.
Let's start with the freebies, available for the download. The most well-know is Paintshop Pro. When I say free, however, be careful. You can download it for free, but it has a limitation on how many times you can use it or for how long. After that, you need to buy it.
There is a considerable price level jump up to the professional level graphics editors like Adobe Photoshop and Corel Photopaint. But it is well worth it. These packages are feature-rich. If you would like to create your own animations, however, lean towards Corel. Adobe Photoshop enjoys a wider user-base, but does not currently do animations. I do not mean to slight Macromedia products, which also have a good following. But to be honest, they tend to be the most expensive.
These products now require you to be running Windows 95 and preferably at least 32 megabytes of RAM, so watch that! If your system is not up to par, you can consider older versions of these products. However, Corel 5, for example, cannot work with JPG and GIF formats! (They are now on Corel 9).
Using the graphics editor, you can crop your photograph and you can introduce special effects if you would like to get creative. You should experiment with these effects; they can be fun. They can also help you to develop totally unique graphics. But they can also create graphics that suck (there IS a web site called WebPagesThatSuck.com, and I recommend you visit).
You may have noticed that some graphics seem to float on the surface of the page and you can "peep" through them. This is the case, for example, with the NED logo on the NED's home page. That effect is called transparency. There is only one file format which permits this - and that is GIF - and not only GIF but a special version of GIF - 89a. Your software lets you specify a color which should be interpreted as transparent, provided you specify this format when saving.
Some photos are large and seem to come in slowly, so you can see progress. This is called interlaced. The interlaced option is available when you save in GIF 87 format.
On the other hand, many experts claim that photos (as opposed to drawings created in Paint programs) are of better quality when stored as JPG files. You will have to be the judge. There have been times when I have converted to GIF formatted from JPG format, and the skin on the people pictured becomes nice and smooth.

And then there are those animations. - the whirling barber pole, for example, or the dancers above. This is achieved by storing a number of frames (at least 18 of them usually) and then assembling them into one file. That file can be stored as an AVI or a GIF or a MOV file. There is a free animator, though documentation for it is scant. You can check for that at my site for downloads. It is called GIF Construction Set.
As you plan out your page, keep in mind that you want your page to be attractive, appealing, enticing, and fast. You want to give your visitors reasons for coming back. If you overload with animations, they will not be impressed as they wait for the page to download to their computer - and they may leave your site before they get a chance to read your stuff! A general rule of thumb is that you should not have more than a couple of animations on any page. Even small animations can be annoying when you have ten of those little guys running around the place. They can be very distracting and annoying, so keep the number of them down.