Here's the boot's backstory, per the eBay auction:
Back when production on the original Batman began, costume designer Bob Ringwood faced a small dilemma. Warner Brothers then had a relationship with shoemaker Nike and asked the designer if there was some way of working the brand into the costuming for the film. Ringwood doubted it, noting that "80s sportswear isn't going to fit in with our 1940s look"... but then inspiration struck, and Nike was tasked with providing Batman's boots for the film. Since the boot was covered by the suit's ankle and shin armor, the Nike shoes became almost unrecognizable in the finished movie.
For 1989's Batman, Nike used the Air Trainer III. Both Michael Keaton and the stuntmen loved the shoe, which provided a comfortable aspect to an otherwise uncomfortable garb. The relationship was so successful that Nike again provided the shoes for the character's second outing in Batman Returns. This time around, Nike used a modified version of their Air Jordan VI shoe, adding an attached boot collar allowing for placement of Batman's shin armor.
The auction is only for a single boot, but that hasn't stopped the current asking price from rising over $2,500 as of this writing.
This piece of cinematic history is available for four more days.