Screen capture is useful to create short instruction videos and the like. GIF is the most common output format. But GIF is an awful format. Consider first creating a normal video first, then using something like gifski to encode to GIF without losing too much quality.

• Peek, animated GIF recorder. Linux, FOSS (repo).
• Gifine, record from a directory of frames, or select a region of your desktop. Linux, FOSS (repo).
• SilentCast, silent mkv screencast and animated GIFs. Linux, FOSS (repo).
• Byzanz, part of GNOME. Linux, FOSS.
• ScreenToGif, screen, webcam and sketchboard recorder. Windows only, FOSS (repo, ghacks article)
• Cropper, screen capture utility. Windows only, FOSS.
• GifCam. Windows only. ghacks article.
• GiftedMotion (Java JAR, very old software, ghacks article)
• LICEcap. Windows, OSX and WINE. Github repo.
• On Mac OS X, there's a built-in QuickTime screen recorder.
• OS X screencast to animated GIF using only free tools, instructions here.
• GIF Brewery. MacOS, Mac App Store.
• Kazam, project seems a bit outdated. Recently suggested by Darren Goossens.

Similar lists by others

• https://www.nextofwindows.com/5-free-tools-to-screen-capture-to-gif-on-windows
• https://itsfoss.com/best-gif-recorder-linux/