ICO is the Windows icon container, and still what a browser fetches when it asks a site for /favicon.ico. One file can hold several sizes at once so the system can pick the right one; each dimension is stored in a single byte, which caps an icon at 256 pixels.
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 an ICO file above and choose afterwards.