On these pages we look at concrete categorories. One definition of concrete categorories is a set together with some form of 'structure' such as functions and operations within the category. 
Some of the simplest concrete categorories are:
Often in category theory we try to keep things as general as possible by working in terms of functor categories where we don't specify the internal structure
Finite Categories
Name  Diagram  Description 

0  No objects No arrows 

1  One object For instance: if the object is 'set' then the whole thing is a monoid. 

2  Two objects here are some examples: 

1+1 that is the disjoint union of two single objects  
This could be an index category  
This could be a graph  
3  Three objects. 
Here we avoid loops (exept identity) which would mean that something else is going on (such as isomorphism or adjunctions).
Categories with binary operation and identity element
single objects  multiple objects >  
composition law  no composition law  
invertible  Groups 
all morphisms are isomorphisms 
Graph  
noninvertible  Monoids  Categories  Directed Graph 