You can download a program called LiceCAP which simply captures the targeted area of the screen, and makes it a GIF.
However you need to use another program like Photoshop or GIMP to resize and edit the GIF if you want to do that.
Also LiceCAP has very awkward quality problems. Some frame rates are inconsistent and jumpy and it has a lower max amount of colours (can't remember the technical name).
If you need someone to capture a video in GIF format there's a good thread in the Premium section where some users will happily and nicely do it if you yourself ask nicely.
If you want a fancy edited one then the GFX section is your best bet.