Smaller GIF image files save space, move fast and could save you a fair bit of money
This jester weighed in at the starting line as a 65.7k JPEG file.

Photoshop Save for Web reduced my JPEG jester to 16k so I could show him here without too much strain on your bandwidth
The picture needed to be even smaller to use as a screen saver or operator logo on a phone, or to send by MMS. Without reducing the pixel size, I Saved it for Web as a GIF at 5.7k by throwing out all but 15 colours.

As you see, I didn't lose too much quality - and I'd reduced the file size by 60 kilobytes
Note. Try this with a photo and you'll find it curdles the colours, giving it a rather crude, posterized look. JPEGs preserve the subtleties of shading. GIFs work best for drawings with areas of flat colour.
You can cut the number of colours in GIF computer images - often quite drastically - and with them file size. Your computer image still looks good, and avoids that machine-turned look you get with a vector image.
The very best tool for this purpose is Photoshop (or Photoshop Elements, miles cheaper, sometimes bundled free with Wacom tablets, and surprisingly powerful). If you're lucky enough to have Photoshop, you simply choose to save your computer image with Save for Web . This opens up a GIF optimiser I'd have given my ears for in the days when I was making banners!
Of course you can do quite a lot towards making your image small with the content of your picture. When you have it to your liking...
PRE-EDIT
Before you enter the 'Save For Web' departure lounge, do some Stage One downsizing in the main screen.
CROP
Can you crop your picture? If so, do it now with the crop tool. Make sure the proportions are about the same as your chosen format.
...Removing the background from a portrait can also save file space, as well as highlighting your subject.
If you want the background to be transparent, fill it now with a colour that doesn't appear anywhere else in your picture. When you save for the web as a .gif, you'll be able to replace that colour with see-through pixels.
Does your GIF have a lot of upright lines? Can you replace them with horizontal lines? Find out the secret of the butcher's apron! Don't despise the little tricks when you're pulling out all the stops to downsize a gif image.


SIZE
Next, go to your Image drop down menu - Image Size. (There's a pixel size reducer in the Save for Web screen, but you have more control if you reduce it here.) Choose your pixel dimensions.
Make sure you have 'Constrain proportions' selected if it's a portrait, or your subject may look like a reflection in a fairground mirror! If one side is still too short, go to Canvas Size and add a bit on, rather than distorting your favourite face.

Be sure to keep your original graphic and save the edited version under a new name. Then you'll always be able to revert if you decide your downsize was a bit too drastic.
Downsize GIF images dramatically in Photoshop