Comparison of Model Category and Set Category
Model categories are introduced on page here.
Set Category  Model Category  

ObjectsIn category theory set objects are often thought of as 'n' elements without any further structure (just an unconnected set of points) or empty set. In model theory objects are spaces, say a topological space with a subobject structure. 

MorphismsIn both cases mappings can be decomposed:


For more comparison of model category and set category see page here.
Mappings in Set
In order to get some sort of intuitive understanding it may help to compare with the following mappings in set:
 bijections
 surjections
 inclusions
These have some interesting properties (wiki):
 Mappings/functions can be decomposed into surjections and inclusions. So we can define mappings as a sequence of surjections and inclusions.
 A bijection is both a surjection and an inclusion.
 A bijection is invertible onetoone correspondence (permutation).
 An inclusion is like a onetoone correspondence with possible additional elements in the codomain.
Mappings and Subsets
Inclusions give us a subset structure. The combination of subobjects and functions from and to them gives us some interesting issues.
For example, in some programming languages such, as Java, we can have functions from and to subobjects (that is an object that has been extended):
CovarianceGiven an object A which has been extended to X. So the map i:A>X is injective. Any function out of X can also be used out of A because we know how all the elements of A are to be treated because they are also elements of X. 
ContravarianceAny function in to A can also be used in to X because we know how all the elements of A exist in X. 
Mappings in Model Categories
Having looked at the above structures in set we can now look at related structures in model categories.
The weak equivalences are like ‘homotopy equivalences’ that is shapes that can be deformed into each other. They are 'weak' because they are spaces that can be represented by simplicial sets, that is there are no awkward singularities or things like that which may be possible in more general case (see: Warsaw circle  ncatlab).
The fibrations are Serre fibrations and play the role of ‘nice surjections’. For a general discussion about fibrations see the page here.
The cofibrations play the role of ‘nice inclusions’.
Retractions
A retraction is a surjective map that is continuous. For instance we can continuously reduce a band down to a circle: 
In general a map ƒ: M1 > M2 is continuous if and only if the preimage of every open set is open. 