What does it mean for a ring to be unital? What is the category of unital rings like? I only know that it no more has a terminal object. But what about the products and coproducts? Are they as usual, different or nonexistent?
In Gelfand theory, a unital C-star algebra means the associated space is compact. Does general unitality have implications on the corresponding scheme?
Thanks
 A: The terminal object is the zero ring. The initial object is $\mathbb{Z}$. Products are cartesian products. Coproducts are free products. Having a unit can be thought of as a kind of compactness condition but strictly speaking, as Zhen Lin says in the comments, in scheme theory all of the rings involved are assumed to have units. 
One way to understand the relationship between unital rings and nonunital rings is the following: the category of nonunital commutative rings is equivalent to the category of augmented unital commutative rings, where an augmented commutative ring is a commutative ring $R$ equipped with a map $R \to \mathbb{Z}$. The equivalence sends an augmented commutative ring $R \to \mathbb{Z}$ to its kernel in one direction and sends a nonunital commutative ring to its unitalization in the other, equipped with a certain natural augmentation.
Passing to opposite categories, we get that the opposite of the category of nonunital commutative rings is equivalent to the category of affine schemes equipped with a map from $\text{Spec } \mathbb{Z}$. Roughly speaking these can be thought of as "pointed affine schemes," in the same way that the opposite of the category of nonunital C*-algebras is equivalent to the category of pointed compact Hausdorff spaces. 
