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 
Illustrating Category Constructions
In order to get an intuitive understanding of category theoretic constructions I find it helps to use examples from the simplest concrete categories to show what is going on internally, especially if it can be illustrated graphically. Of course, we are not really supposed to look inside categories, but I think its necessary to get an intuitive understanding.
Often sets are the simplest category to show whats going on.
If sets don't have enough structure the a preorder (see this page) is often a good example to use (on this page I have tried to use preorders to show the idea of adjunctions). preorders are good because of the graphical nature of Hasse diagrams.
Preorders are also good because they are a specific (less general) case of categories in that, like categories, they are reflexive and transitive:
reflexive 
a≤a If a relationship is reflexive then there is always an arrow from an element of a set to itself. Although when drawing the Hasse diagram we may not draw it to avoid clutter. 
transitive 
if a≤b and b≤c then a≤c This is like composing functions. That is: if the is an arrow from 'a' to 'b' and an arrow from 'b' to 'c', then we can attach the tip of the first arrow to the second arrow to get an arrow from 'a' to 'c'. 