"Comma categories were introduced by Lawvere [1963] in the context of the interdefinabilty of the universal concepts of category theory. The basic idea is the elevation of arrows of one category to objects in another"  Computational Category Theory, D.E. Rydeheard R.M. Burstall. 
Slice category is a specific case of comma category. To introduce the subject gradually lets start with the slice category then move on to the more complicated comma category later.
Slice Category
Given a category C we can 'slice' it over some object XC which we fix in C.
This construction allows us to start with one category 'C' and generate a different category 'C/X' by fixing a given element 'X' in C. The elements in C/X are pairs (A,P) where AC and P is a morphism from A to X. 
'coslice' is the dual concept, just reverse all the arrows.
Slice Category Examples
On the page here I have put more information about these examples
Colouring of labeled set. Here we choose as our fixed object 'X' the 3 element set containing R, G and B (for red green and blue). Every object (in this case set) has an arrow (function) to this set so all the elements are assigned a colour. 

Adding a distinguished point (the origin) to Euclidean space to give a vector space. 
Comma Category
In the slice category all the objects are in the same category (including the fixed object). The comma category generalises this by allowing the objects to come from other categories. 
Special Cases of Comma Category
We have already seen the slice category, here is a fuller list of special cases and related constructs:
Slice CategoryIf 's' is the identity functor of C and 't' is the inclusion 1 > C of an object cC, then (s/t) is the slice category C/c. See ncatlab site. 

Coslice CategoryLikewise if 't' is the identity and 's' is the inclusion of c, then (t/s) is the coslice category c/C See ncatlab site. 

Arrow CategoryThis is all derived from a single category. So it is a comma category where the arrows into it are identity functions. Unlike the slice or coslice both the source and target can be somthing other than the terminal object.


Graph CategoryThe category of graphs is an example of a comma category, that is, a graph is isomorphic to a comma category. Arrows of graphs are pairs of functions mapping nodes to nodes (N > N) and edges to edges (E > E) , an object of this category it a triple (E,f: E > N×N,N) where f maps each edge to a source and target node. See page here. 

Universal Morphism"The notion of a universal morphism to a particular colimit, or from a limit, can be expressed in terms of a comma category. Essentially, we create a category whose objects are cones, and where the limiting cone is a terminal object; then, each universal morphism for the limit is just the morphism to the terminal object. " academickids.com See page here. 

Adjunction"Lawvere showed that the functors F : C>D and G : D>C are adjoint if and only if the comma categories FD and $C$G $are\; isomorphic,\; and\; equivalent\; elements\; in\; the\; comma\; category\; can\; be\; projected\; onto\; the\; same\; element\; of\; C\times D.\; This\; allows\; adjunctions\; to\; be\; described\; without\; involving\; sets,\; and\; was\; in\; fact\; the\; original\; motivation\; for\; introducing\; comma\; categories."academickids.com$ See page here. 

Kan ExtensionSee page here. 
Discussion
The comma category is a specialisation of the arrow category where the codomain is the same for all objects.  
Alternatively there is also a cocomma category where the domain is the same for all objects.  
The above diagram could be rewritten with the X shown in each object to make it look more like the arrow category. 
The concept of a comma category is related to the idea of a fibre and sheaf, reading these pages for may help with intuition.
Then the comma category CX has, 
The cocomma category XC has, 

Objects 

pairs(A,f)



Morphisms 




Terminal Objects in Comma 
Initial Objects in CoComma 

If C has a terminal object '1' exists then: C/1 = C 
If C has an initial object '0' exists then: 0/C = C 

Products in Comma 
CoProducts in CoComma 

In terms of the above 'pairs' we can construct a product:
We can reduce the above diagram to: 
General Comma Category
The comma category can be generalised further by making the target also a functor. 
Further Genralisation
So we fix f: C>D and XD
Then the slice category C/X has, 
The coslice category X/C has, 

Objects 




Morphisms 



We can generalise this further by not fixing X
Instead x is combined with the pair to give a triple.
Then the comma category FX has, 
The cocomma category FC has, 

Objects 

triple(cC,xD,p)



Morphisms 


Further generalisation, we add a third category 'E' like this:
Then the comma category FX has, 
The cocomma category FC has, 

Objects 



Morphisms 


Comma Category References
 TheCatsters  Youtube  Slice categories 1 Nov 2008
 TheCatsters  Youtube  comma categories 2
 Wiki
 nlab
 Richard Southwell  Universal Properties and Comma Categories  Youtube video  51 min into video
 Richard Southwell  Adjoint Functors  Youtube video 30 min into video