Short answers, pulled from the story.
A set in mathematics is a collection of different things called elements or members. Those elements are typically mathematical objects such as numbers, symbols, points in space, lines, geometric shapes, variables, functions, or even other sets.
Georg Cantor, who lived from 1845 to 1918, began the mathematical study of infinite sets. His work showed that the number line has strictly more elements than the natural numbers and that any line segment has the same number of elements as the whole line.
Since the first half of the 20th century, ZFC, meaning Zermelo-Fraenkel set theory with the axiom of choice, has been the most commonly used axiom system. It was generally adopted as the foundation of set theory and all mathematics after the foundational crisis.
The continuum hypothesis is a conjecture formulated by Georg Cantor in 1878 stating that there is no set with cardinality strictly between that of the natural numbers and that of the real numbers. In 1963, Paul Cohen proved it is independent of the axioms of Zermelo-Fraenkel set theory with the axiom of choice.
The axiom of choice says that given any family of nonempty sets, one can choose an element from each of them simultaneously. A more formal version states that the Cartesian product of every indexed family of nonempty sets is nonempty, and it is equivalent to Zorn's lemma and to well-ordering.
Countable sets are those with cardinality less than or equal to the cardinality of the natural numbers, including finite sets and countably infinite sets. Uncountable sets are those with cardinality strictly greater than that of the natural numbers, as shown by Cantor's diagonal argument.