Q:

The differences between the following conception

Till now I've not understood the differences between ( function ) , (mapping ) , (application) , 

A:

Let's start with function.  A function from one set $R$ to another $D$ (where possibly $D$ is the same as $R$ is a binary relation between elements of $R$ and elements of $D$, such that for very element of $R$ the relation specifies exactly one element of $D$. This can be thought of as a set of pairs $r_i, d_i$ such that every $r \in R$ appears in exactly one of those pairs.
A mapping, or map, is almost synonymous with function, but in specific branches of mathematics, "map" is defined to be a function with some additional property. For example, in topology, a map is a  continuous function $f$ (a function such that for any given open set $y\subset D$, the pre-image of Y which is 
$\{ x\in R : f(x) \in Y \}$ is an open set in $R$.
In addition, in computer science and occasionally mathematics, one might talk about a one-to-many map.
Application of a map or function is the act of going from $r \in R$ to $d \in D$ according to the rules of that map or function.  

