What Does it Really Mean to Have Different Kinds of Infinities? Can someone explain to me how there can be different kinds of infinities?

I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me.

Any help would be appreciated.
Suppose no one ever taught you the names for ordinary numbers.  Then suppose that you and I agreed that we would trade one bushel of corn for each of my sheep.  But there's a problem, we don't know how to count the bushels or the sheep!  So what do we do?

We form a "bijection" between the two sets.  That's just fancy language for saying you pair things up by putting one bushel next to each of the sheep.  When we're done we swap.  We've just proved that the number of sheep is the same as the number of bushels without actually counting.

We can try doing the same thing with infinite sets.  So suppose you have the set of positive integers and I have the set of rational numbers and you want to trade me one positive integer for each of my rationals.  Can you do so in a way that gets all of my rational numbers?

Perhaps surprisingly the answer is yes!  You make the rational numbers into a big square grid with the numerator and denominators as the two coordinates.  Then you start placing your "bushels" along diagonals of increasing size, see wikipedia.

This says that the rational numbers are "countable" that is you can find a clever way to count them off in the above fashion.

The remarkable fact is that for the real numbers there's no way at all to count them off in this way.  No matter how clever you are you won't be able to scam me out of all of my real numbers by placing a natural number next to each of them.  The proof of that is Cantor's clever "diagonal argument."
Just simple intuitive explanation.

How many Natural (Integers) Numbers could you count?
There are infinitely many, yet you can count them.
It's called Countable Set.

How many Real Numbers are there?
Infinitely as well (Since at least every Natural Number is a Real Number).
Yet you won't be able to count them (Intuitively, Let's say you name a number the first, then find the second, I can, for sure, find a number in between, their average which is Real Number as well). 
It's called Uncountable Set.

What you are after is how we define how big is a given set.
Then you should look for Cardinality.