Product (pullback) 


generalisation  a kind of limit  
set example  cartesian product {a,b,c}*{x,y}= 

group  the product is given by the cartesian product with multiplication defined componentwise.  
Grp (abelian)  direct sum  
vector space  direct sum  
poset  greatest lower bound meet 

base topological space  
POS  greatest lower bounds (meets) 

Rng  
Top  the space whose underlying set is the cartesian product and which carries the product topology  
Grf  
category  objects: (a,b) morphism: (a,b)>(a',b') 
tensor products are not categorial products.
In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).
Sum
When generating a sum for objects with structure then the structure associated with the link can be added to the sum object.
Product
Products for groups are discussed on this page.