A pushout in category theory is a kind of colimit. This is a generalisation of a sum discussed on page here. It consists of the sum A+B with two arrows into it, one from A and the other from B. For the pushout we add a third object C with arrows into A and B such that the square commutes. 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. 
Example in Set
Here we have added set C to the diagram (on the page about sum). Now the square needs to commute. 
Example in Directed Graph
How does this work when we add structure to the set? For instance, in a directed graph, can we have edges into and out of the intersection? 
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.