Screen to Gif is a free portable tool which allows you to record desktop activity, edit it, then save the results as an animated GIF.
The program opens with a transparent resizable frame which you must position and resize around the area you'd like to record. You're also able to define how many frames per second your animation will use. (By default this is set to 15, although you can increase it to 40 if it's really necessary.)
Click Record (or press F7), then launch programs, open files or do whatever else you like, and Screen to Gif will record whatever happens within its frame. You can pause and resume this as required, with a click or a (customisable) hotkey.
Click Stop when you've finished and your recording appears in an editor. Your first steps here might be to trim it down to size, deleting unwanted sections from the start or end of the recording, and removing individual frames wherever you like.
You can then tweak what's left in a variety of ways, perhaps resizing or cropping your footage, slowing it down, inserting new frames, adding text captions, even applying a range of filters to one or all frames (Greyscale, Blur, Negative, Pixelate, Colour, Sepia).
Click Done when you're finished and Screen to Gif saves the results as an animated GIF, then offers to open the file for you.
Version 1.4 is a bug fix release.