What's the difference between "relation", "mapping", and "function"? I think that a mapping and function are the same; there's only a difference between a mapping and relation. But I'm confused. What's the difference between a relation and a mapping and a function? 

Mathematically speaking, a mapping and a function are the same. We called the relation 
$$
f=\{(x,y)\in X\times Y : \text{For all $x$ there exists a unique $y$ such that $(x,y)\in f$} \}
$$
a function from $X$ to $Y$, denoted by $f:X\to Y$. A mapping is just another word for a function, i.e. a relation that pairs exactly one element of $Y$ to each element of $X$.
In practice, sometime one word is preferred over another, depending on the context.
The word mapping is usually used when we want to view $f:X\to Y$ as a transformation of one object to another. For instance, a linear mapping $T:V \to W$ signifies that we want to view $T$ as a transformation of $v\in V$ to the vector $Tv\in W$. Another example is a conformal map, which transforms a domain in $\Bbb C$ to another domain.
The word function is used more often and in various contexts. For example, when we want to view $f:X\to Y$ as a graph in $X\times Y$.   
