The game stars an Adam West-looking Batman on a scavenger hunt around the deadly Batcave for seven pieces of the Batcraft (so he can get it hovering again and go rescue Robin, of course). According to the Sinclair User review (issue #50, pg. 51), Batman is notable for being an early, if not the first, game to implement a checkpoint system.
The unofficial remake of Batman was developed by Tomaz Kac (who also faithfully remade Ritman and Drummond's Head Over Heels followup) and is available to download and play for free on PC. An earlier freeware remake, dubbed Watman, was released for DOS and Game Boy Advance about ten years ago. Consider yourself schooled.
[Image sources: Retrospec / RudyKoch]