I would guess it's a laminate counter (e.g. Formica), a thin layer of a plastic sort of material on a substrate of plywood or particle board or similar.
You should be able to find laminate materials at most building centers/big box home improvement stores. They are not too hard to install on flat surfaces (using a contact adhesive). Square corners are also pretty easy to do, with trimming router bits to square up the edges of the laminate, but curves other than quite gentle ones are rather impractical for DIY installation (at least to my understanding).
You could also replace the whole counter top, rather than covering it with a laminate; there are many options available. Some can get to be rather heavy if the island is large.