A sum in category theory is a kind of colimit. It is dual to a product discussed on the page here. It consists of the sum A+B with two arrows into it, one from A and the other from B. It must have a universal property which is: For any other object Z with maps from A and B there must be a unique arrow from A+B to Z. 
Sum Example in Set
In this diagram the two sets A and B are shown with elements. Some of the elements in A+B have inputs from A, some from B and some from both. So A+B is a (not necessarily disjoint) union. So to comply with the universal property the maps into Z must respect the disjointedness or otherwise of the elements. 
I have drawn the sum like this:  
but when encoding the elements we may need to tag them with a 0 or 1 to know whether they come from A or B. In algebraic terms this is like: A+A = 2*A where 2 means a Boolean type 
(a,0) (b,0) (c,0) (d,0) (c,1) (d,1) (e,1) (f,1) 
Example in Directed Graph
This is like the set example with arrows between the elements. Can we have arrows into and out of the intersection like this? I can't see why not but possibly not when we go on to pushout. 
Table of Results
Sum 


generalisation  a kind of colimit  
set example  disjoint union {a,b,c}+{x,y}= 

group  free product the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute. 

Grp (abelian)  direct sum the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero) 

vector space  direct sum  
poset  least upper bound join 

base topological space  wedge  
POS 

least upper bounds (joins) 
Rng  
Top  disjoint unions with their disjoint union topologies  
Grf  
category 
Sum
When generating a sum for objects with structure then the structure associated with the link can be added to the sum object.
Next Steps
see also: