That is, a category C is complete if every diagram F : J → C (where J is small) has a limit in C. Dually, a cocomplete category is one in which all small colimits exist.
The existence of all limits (even when J is a proper class) is too strong to be practically relevant.
It follows from the existence theorem for limits that a category is complete if and only if it has equalizers (of all pairs of morphisms) and all (small) products.
[1] A small complete category is necessarily thin.
Without the finiteness restriction a posetal category with all products is automatically cocomplete, and dually, by a theorem about complete lattices.