This is difficult to answer because you do not clearly show your render settings.
While an animated GIF may showcase your directorial acumen, it is not very informative.
That said, be sure to check if you have compositing enabled, if you're not using it then disable it.
If you are using it then check that the compositor nodes are wired up. A lot of users have the issue that the output node is disconnected, resulting in the grey screen of render. Not sure if this is a default setting in blender, if so it should probably be listed as a bug as it regularly appears here as a question.
If that doesn't work, I suggest you use the search facility of this site to track down your problem:
Search this site for render issues with grey output