Ed and Aaron, thank you very much for explaining the practical benefits of injectivity and surjectivity.
I have taken what I learned from you and cast it into my own thoughts and experiences. Please let me know of any errors.
Motivation
When I go hiking, I want to be able to retrace my steps and get back to my starting point.
When I go to a store, I want to be able to return home.
When I swim out from the shore, I want to be able to get back to the shore.
Going somewhere and then coming back to where one started is important in life.
And it is important in mathematics.
And it is important in functional programming.
Domain, Codomain, and Inverse
If a function maps a set of elements (the domain) to a set of values (the codomain) then it is often useful to have another function that can take the elements in the codomain and send them back to their original domain values. The latter is called the inverse function.
In order for a function to have an inverse function, it must possess two important properties, which I explain now.
The Injective Property
Let the domain be the set of days of the week. In Haskell one can create the set using a data type definition such as this:
data Day = Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday
Let the codomain be the set of breakfasts. One can create this set using a data type definition such as this:
data Breakfast = Eggs | Cereal | Toast | Oatmeal | Pastry | Ham | Grits | Sausage
Now I create a function that maps each element of the domain to a value in the codomain.
Here is one such function. The first line is the function signature and the following lines is the function definition:
f :: Day -> Breakfast
f Monday = Eggs
f Tuesday = Cereal
f Wednesday = Toast
f Thursday = Oatmeal
f Friday = Pastry
f Saturday = Ham
f Sunday = Grits
An important thing to observe about the function is that no two elements in the domain map to the same codomain value. This function is called an injective function.
[Definition] An injective function is one such that no two elements in the domain map to the same value in the codomain.
Contrast with the following function, where two elements from the domain -- Monday and Tuesday -- both map to the same codomain value -- Eggs.
g :: Day -> Breakfast
g Monday = Eggs
g Tuesday = Eggs
g Wednesday = Toast
g Thursday = Oatmeal
g Friday = Pastry
g Saturday = Ham
g Sunday = Grits
The function is not injective.
Can you see a problem with creating an inverse function for g :: Day -> Breakfast?
Specifically, what would an inverse function do with Eggs? Map it to Monday? Or map it to Tuesday? That is a problem.
[Important] If a function does not have the injective property then it cannot have an inverse function.
In other words, I can't find my way back home.
The Surjective Property
There is a second property that a function must possess in order for it to have an inverse function. I explain that next.
Did you notice in the codomain that there are 8 values:
data Breakfast = Eggs | Cereal | Toast | Oatmeal | Pastry | Ham | Grits | Sausage
So there are more values in the codomain than in the domain.
In function f :: Day -> Breakfast there is no domain element that mapped to the codomain value Sausage.
So what would an inverse function do with Sausage? Map it to Monday? Tuesday? What?
The function is not surjective.
[Definition] A surjective function is one such that for each element in the codomain there is at least one element in the domain that maps to it.
[Important] If a function does not have the surjective property, then it does not have an inverse function.
[Important] In order for a function to have an inverse function, it must be both injective and surjective.
Injective + Surjective = Bijective
One final piece of terminology: a function that is both injective and surjective is said to be bijective. So, in order for a function to have an inverse function, it must be bijective.
Recap
If you want to be able to come back home after your function has taken you somewhere, then design your function to possess the properties of injectivity and surjectivity.
If $f$ is a function $A\to A$ where $A$ is a finite set, then $f$ is injective if and only it it is surjective. It is not possible for such an $f$ to be injective without being surjective or vice versa.
This can be proved by induction on the number of elements in $A$, but the details are slightly tedious. This property carries over to a function $f:A\to B$ where $A$ and $B$ have the same number of elements -- because "the same number of elements" means that there is a bijection $g:A\to B$, and then $f$ is injective/surjective exactly when $g^{-1}\circ f$ is, and $g^{-1}\circ f$ goes from $A$ to $A$ itself.
On the other hand, if $A$ is infinite, then it is possible for a function to be injective without being surjective, or vice versa. Some simple examples for $\mathbb N\to\mathbb N$ is
$$ g(n) = n+7 \\
h(n) = \begin{cases} n & \text{if }n\le 42 \\ n-1 & \text{otherwise} \end{cases}$$
where $g$ is injective but not surjective, and $h$ is surjective but not injective.
(Strictly speaking this property of infinite sets is only guaranteed if the axiom of choice holds, but that is a techincal complication that you probably don't want to worry about at this level).
Best Answer
Let $X,Y$ be sets and $f:X\rightarrow Y$ a function. The idea of a function is, that every $x\in X$ is mapped to only one $y\in Y$. If $X$ and $Y$ are now finite (as you are looking for, right?), then there are no functions which can fulfill 1.) or 2.).
Maybe this will help for better understanding. Try to understand for $|X|=|Y|$:
If $f$ is surjective then it is also injective (thus bijective) and vice versa.