This may not be an exact representation of what goes on in a GIF, but I'm guessing it's close enough for our purposes.
Let's suppose you have a 6x6 gif image made up of 3 colors a,b and c...
Or in terms of a, b and c...
c c c c c c c b b b b b a = red c b b b b b b = green c a a b b b c = blue c a a b b b c c c c c c
One way to "describe" this image is
first row c-c-c-c-c-c second row c-b-b-b-b-b etc
Each pixel is defined one at a time.
Let's suppose we tried to save a little time though and described it like this...
1:6c 2:c-5b etc
By attempting to group pixels together you save room. If we had a very large image that started off like this...
first row c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c second row c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c-c
A gif might simply say...
1:25c 2:25c
This is gif "compression".
Let's suppose that instead of all c's in those first two rows, there are other colors...
first row c-c-c-c-b-b-b-a-b-b-a-a-a-a-a-c-c-b-b-b-a-b-c-c-c second row a-a-a-a-b-b-b-a-b-c-c-c-c-c-b-b-b-b-c-c-c-c-c-c-c
This would compress into...
1:4c-3b-a-2b-5a-2c-3b-a-b3c 2:4a-3b-a-b-5c-4b-7c
Now let's suppose instead of 3 colors, we reduced the number of colors to 2. We'll remove a and substitute b instead. If a and b are similar colors, the image quality may not suffer very much at all...
first row c-c-c-c-b-b-b-b-b-b-b-b-b-b-b-c-c-b-b-b-b-b-c-c-c second row b-b-b-b-b-b-b-b-b-c-c-c-c-c-b-b-b-b-c-c-c-c-c-c-c
This would compress into...
1:4c-11b-2c-5b-3c 2:9b-5c-4b-7c
Compare this to the last example. Do you see how this would translate into a smaller file size?
One of the worst things you can do to a gif is make it speckled...
first row c-b-c-b-c-b-c-b-c-b-c-b-c-b
Because what would that compress into?
1:c-b-c-b-c-b-c-b-c-b-c-b-c-b
You don't save a thing.
So, the whole business of gif compression revolves around one simple idea... reduce the number of colors and/or color changes in the file.
Related articles you may find interesting... GIF or JPG and Image Compression Examples.