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?

Good question. I can give you a simple example.

You can generally think of relations as functions in a higher dimensional space, intersected with a zero plane.

So for example, lets take a parabola and a circle to demonstrate this. Imagine the parabola

$$ f(x) = x^2 + 3x $$

This is clearly a function of x, because if you give me an x value, I can give
you the corresponding value of f(x) - a mapping is really just another name
for a function. If we want to graph it, we can let the y value
equal the output of $f$, so we would get this graph:

On the other hand, if we graph a circle, like:

$$x^2+y^2=4$$

Its graph is given by:

Now this is fundamentally different to the function. If you wanted the y value at x = 0,
I can't point to a unique value, I'd have to say y = 2 or y = -2. A function can only
have one output. So we have to call this a relation.

Higher Dimensions

However, we can use a clever trick for this circle. We can rewrite it as:

$$ x^2 + y^2 - 4 = 0 $$

Which is obviously the same thing, but on the left hand side, notice that we now
have a function of (x,y), so we can think of this like:

$$ g(x, y) = 0 $$

In a higher dimension, this would be the intersection between the shapes:

$$ z = g(x, y) $$

and

$$ z = 0 $$

Which I've shown below:

Notice that same circle hiding in plain sight.

Key takeaway (tl;dr)

Relations are functions in a higher dimension, intersected with a zero plane
in the higher dimension.

Mathematically speaking, a mapping and a function are the same. We called the relation
$$
f={(x,y)in Xtimes Y : text{For all $x$ there exists a unique $y$ such that $(x,y)in f$} }
$$
a function from $X$ to $Y$, denoted by $f:Xto Y$. 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 $f:Xto Y$ 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 $vin V$ to the vector $Tvin W$. 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 $f:Xto Y$ as a graph in $Xtimes Y$.

There is basically no difference between mapping and function. In algebra, one uses the notion of operation which is the same as mapping or function. The notion of relation is more general. Functions are specific relations (those which are left-total and right-unique).

Relation and Function are quite different as the later only consider unique images.

There's not much difference between a mapping and function. The only difference that I can think of is that mapping is just the diagramatical representation of a function or an arrow diagram representation of a function which is quite clear for functions defined on the finite sets, while for infinite sets the two notions coincide.

I Hope it Helps...