First let us consider an example.
Consider the following order $\prec$ on $\mathbb N$:
$m\prec n\iff\begin{cases} m\text{ is even},\ n\text{ is odd}\\ m,n\text{ are even },\ m
In simple terms we took all the odd numbers and declared them larger than all the even numbers, but between two even numbers (or two odd numbers) the relation is the same.
To see this is a well-order, first note that it is a (strict) linear order. Every two elements are comparable. Either both have the same parity and we know what to do, or one is odd and one is even and we know what to do.
Now consider $A\subseteq\mathbb N$ non-empty, if it contains even numbers then the smallest even number (in the usual ordering, $\lt$) is the minimal element of $A$. Otherwise $A$ contains only odd numbers, and therefore the minimal (again in the sense of $\lt$) is the minimal element of $A$.
Okay, so we have a well ordering of $\mathbb N$, big whoop. But wait, what is the successor of $0$ in this ordering? What is the number that comes right after $0$? It is no longer $1$, but now it is $2$. On the other hand, how many numbers are smaller (in the sense of $\prec$) than $1$? Well, all the even numbers are, so we have infinitely many of them. The order looks like this:$0,2,4,6,8,10\ldots,1,3,5,7,9,\ldots$
Right, what was that good for? Well, first it helps to understand that well ordering is not necessarily the usual order on the natural numbers. In fact there are so many different ways to well order $\mathbb N$ that we cannot describe each and every one of them. There are uncountably many ways, to be more accurate, to do so (even if we consider two the same if they are order isomorphic).
I think that we can now move to discuss a well order of the real numbers. The real numbers can be well ordered as a trivial consequence of the axiom of choice (well, every set can be well ordered using the axiom of choice and in particular the real numbers). However we cannot describe, as I did with $\prec$ above, any way to well order the real numbers. There are certain situations where we can give a nice definition of such well order, but generally speaking we can only prove the existence of this well order, and indeed we need the axiom of choice to prove this existence.
This is very much like that we can prove that there are numbers which are irrational, but we cannot name too many of them. In fact, most real numbers are transcendental and we cannot even describe them using $+,-,\times,/,\sqrt[n]{}$ and the integers. We can still prove that they exist though.
One last remark on the above, this does not mean that every uncountable set is so complicated that we cannot describe an explicit well ordering of it. There are uncountable sets whose order is rather simple, but this post is already long enough.