A category is a collection of objects of the same kind, together with morphisms between them when defined. The collection of all sets and all functions between them is a category. But there are so many objects of the same kind, that a category is usually not a set. E.g. the set of all sets is too big to be a set, at least in the set theoretic language I use, where "big" collections are called classes and only small ones are called sets.
To me, a variety is a reduced, irreducible, scheme of finite type, usually defined over an algebraically closed field, so is a very special type of scheme. In particular, on every open subset of a variety, the ring of sections of the structure sheaf is not just some arbitrary ring, but an integral domain of k valued functions containing the base field k. (See Mumford's redbook, chapter II.3) These are very close to those closed subsets of projective space defined by homogeneous prime ideals, or (Zariski) open subsets of them, and functions which are restrictions of quotients of homogeneous polynomials.