Most mathematicians intuitively understand the statement "The cardinality of the real numbers is strictly greater than the cardinality of the natural numbers". Of course there are more real numbers than natural numbers. The statement "the cardinality of the rational numbers is the same as the cardinality of the natural numbers" is less intuitive, but after a first course in analysis almost everyone is ready to believe it. However these types of statements and the arguments involved to prove them only allow us to talk about cardinality in relative terms. To talk about the size of a single set, we need to introduce the cardinals, a class of canonical representatives that arise when we identify sets of the same size.
I will start with an explanation of what ordinals and cardinals are, and then discuss what it means for an infinite cardinal to be large (given that we are talking about an increasing proper class of infinite sets, this turns out to be nontrivial). We will go over some examples, touch on the large cardinal hierarchy and some applications, and finish with a discussion about how large cardinals allow us to win games.