Step 1: Get Everything You Need.
Next, you need a program that will allow you to take screenshots from a movie file. There are many programs that do this. The one I'm using is a free program called "Movie Snapshot," which is available here: http://www.cd2html.de/moviesnapshot.en.html
UPDATE: Another program I would reccomend for this task is VirtualDubMod. It's less simple, and a little more daunting than Movie Snapshot, but I also find it to be a little more stable and compatible with opening movie files. It also has many more features than Movie Snapshot. You can get it here: http://virtualdubmod.sourceforge.net/
Last, you need the GIMP, a free Image editing program, which you can get from this website: http://gimp-win.sourceforge.net/
Step 2: Get the Screenshots
It's a good idea to get something that will be short, simple, and loop easilly. I chose three frames of Bart and Lisa screaming from an episode of the Simpsons.
UPDATE: If you'd like to do this step with VirtualDubMod; open it up, then push Control-O to browse for your video file and open it. You can use the arrow keys to advance one frame forward and backward, or use the arrow buttons at the bottom. You can also use the slider bar, to move quickly through the movie. Once you've found the first frame you want to use, push Shift-1 and it will pop up with a little window, asking where you want to save it. VirtualDubMod defaults to saving in the .TGA format, which you really don't want to do. When saving pictures from VirtualDubMod, choose Bitmap (.bmp) as the output format, because it's a format everythign can open, and it doesn't compress the data. After you've saved the first picture, just advance the video by one frame and save again until you got the whole length of the video that you wanted to use, just like normally.
Step 3: Load Up Your Pictures in the GIMP
Now, Open up GIMP, and drag the first image into the main window, loading it. A new window, with the image, should pop up in GIMP. Also, a window entitled "layers" should pop up. If it doesn't, push Ctrl+L to bring it up. There should be one layer, currently, "Background," which contains the image you dragged into GIMP.
From the folder you're keeping them in, drag the rest of your images into the Layers window of GIMP, one at a time.
Step 4: Make It an Animation
Once they're in order, close the animation preview window, and right click on the image, going to the animation menu again, but this time click on "Optimize (For GIF)" It will think for a bit, then come up with a new image window. You can close the old one; we wont be needing it anymore.
Now, click on the new window, and you'll notice that the Layers window changes to reflect the new image. In the layers window, the names of the layers each contain "(100MS)." If you wanted to change the speed your GIF plays as, then do it here, by double-clicking on the name of each layer and changing the 100 to another number. Smaller numbers mean faster; larger numbers mean slower.
You may want to preview the animation again at this point, to make sure everything looks right.
Step 5: Resize the Image
First, you might want to crop the image, to change it's dimmensions. I can make the final image be taller if I cut of the space on the left and right edges so that the image over-all is square.
To do this, right click on the image, then go to Image>Crop That will open up a window like the one shown here. Next to the "Width" and "height" boxes, there is a little chain. Click it so that it is broken. Next, if you want to narrow your image, lower the value in the width box. You'll notice that there's a little thumbnail of your picture that show what is being cut off. Don't like it only cutting off from one side? Then mess with the value in the X:Offset box towards the bottom of the window. Fiddle with this and the Width box, until the thumbnail looks correct. You can also do the same thing with the Height and Y:Offset, if you want to cut off uninsteresting parts from the top and bottom. Click Resize.
Now, we want to resize the image. Right click on the image again, and then go to Image>Resize. Lower the width of height values until you it's their value is within the size restricions you have. In this case, 100X100. Under Interpolation, go to "Cubic" for the best quality. Click on scale.
Step 6: Save It As a GIF
Click export, click ok in the rest of the windows that pop up, and it will save you a brand new animated gif. Now open it, and feel proud of yourself as you watch it.