What type of boat might this be? Or what category of boats would it fall under?
That is a cog,
A cog is a type of ship that first appeared in the 10th century, and was widely used from around the 12th century on. Cogs were clinker-built, generally of oak, which was an abundant timber in the Baltic region of Prussia. This vessel was fitted with a single mast and a square-rigged single sail.
It's a military Cogue/Cog/Koggen with fore and aft archery towers. Cogues are the only medieval single mast square rigged sailboats of this style.