An object Q is couniversal, or terminal, if for every object R in the category, there is one and only one morphism from R to Q.
If Q and R are both initial there is one morphism from each to the other, and the composition has to be the unique morphism from Q into itself, which is the identity morphism. In other words, the two objects are equivalent.
There is only one initial object, and one terminal object, up to equivalence.
A zero object is both initial and terminal.
A category need not have all these objects. For instance, in the category of sets, the empty set is initial, a set with 1 element is terminal, and since these objects have different cardinalities, there is no zero object. If the empty function is not considered a valid morphism, there is no initial object either. Categories such as groups, modules, and rings have a zero object, namely the identity element.