I Googled "ordinal numbers" and stuff for kids came up, on a website I found.
"An Ordinal Number is a number that tells the position of something in a list, such as 1st, 2nd, 3rd, 4th, 5th etc"
So my question is, what do ordinal numbers in this sense have to do with ordinal numbers in Cantor's set theory?
On
To add to the other answer, an ordinal describes the order type of a well-order. A well-order is a total order for which every nonempty subset has a least element.
Everyday ordinals are just finite ordinals. Now, it turns out that every finite total order is a well-order. Therefore, everyday ordinals describe the order type of finite total orders. That is, we take a finite set and line up its elements in some fashion, and an ordinal describes the position of some element in this arrangement.
Ordinal numbers have to do with order (hence the name); which object is first, second, third, and so on depends on what order the objects are arranged in. Likewise, ordinal numbers in the sense of Cantor have to do with the order in which things happen. A good example is the following - consider the following re-ordering of the natural numbers:
$$0,2,4,\ldots, 1,3,5,\ldots$$
This lists the even numbers first and then the odd numbers. It's easy to identify each number with a number indicating its position in this list, at least at first: $0$ is the $1$st element, $2$ is the $2$nd, $4$ is the $3$rd, and so on. But what about $1$? It comes after infinitely many numbers, but it's the first number that does. That makes it the $\omega$th element of the list - $\omega$ being the first infinite ordinal. And then $3$ is the $\omega + 1$st, and so on. So ordinal numbers of whichever sort are numbers which can describe something's location in a list.
By contrast, a cardinal number describes something's size, regardless of order. In English, these are called one, two, three, and so on; these are words which talk about how many things are in a collection, not about what order they're in. If I tell you I have five sheep, I haven't told you anything about how they're arranged. But, as the example above suggests, if I tell you I have $\omega$ sheep, I've accidentally told you something about their order - they can't be arranged like my list above, in two infinite groups, because then I'd have an $\omega + 1$st sheep! So $\omega$ doesn't seem to be the same kind of number as "five". Instead, we use cardinals in the sense of set theory - $\aleph_0$ means "the same quantity as $\omega$, but not in any particular order".