Thirty-five years old and still everywhere, because it does one thing nothing else quite does: animate inline, silently, in any context that displays an image. The cost is a 256-colour palette and no real motion compression, which makes GIF a poor way to store video and an excellent way to send a three-second loop.
Every conversion below runs in this page, on your own machine. Nothing is uploaded, there is no queue, and there is no account to make. Pick a target format, or just drop a GIF file above and choose afterwards.