A bijective function \(f: A\to B\) satisfies the property that for all \(b\in B\), there is exactly one \(a\in A\) such that \(f(a) = b\). So there is a one-one correspondance between the elements of \(A\) and \(B\). Clearly if there is a bijective function \(f\) from \(A\) to \(B\), the function has an inverse \(f^{-1}:B\to A\) that is also a bijection.
By definition, the cardinality of \(A\) and \(B\) are equal if there is a bijection from \(A\) to \(B\). This is how we compare infinite sets.
Definition All finite sets are countable. An infinite set \(A\) is countable iff there is a bijection \(c:A\to {\mathbb N}\), where \({\mathbb N}=\sets{1,2, \ldots\) is the set of all natural numbers.
Your handout gives many examples of countable sets. Informally this means that given an infinite sheet of paper and infinite time, you could write all elements of \(A\) one below another (because you can write natural numbers one below another as a list).
Cantor came up with a stunningly beautiful argument to prove that the set \([0,1]\) is not countable. While both \([0,1]\) and \({\mathbb N}\) have an infinite number of elements, the size of \([0,1]\) is a bigger infinity than the size of \({\mathbb N}\).
This means that even given infinite time, you couldn’t write out all elements of \([0,1]\) one below another in that infinite sheet of paper (in any order). This is not because there isn’t a notion of “the next real number after .1”. In fact, you could write all rational numbers one below another (though there is no “next rational number after .1” either).
There are any number of articles on this on the Internet, here is a short video explaining it. This argument is very fundamental—it is used to prove a lot of other things, for example that the Halting Problem is undecidable (in very imprecise terms, this means there are problems that the Turing model of computation cannot solve). Who knows, you may get to use it in grad school for your research too, as one of my students did to show that some models cannot be infered well no matter how much data you see.