Short answers, pulled from the story.
Infinity is something boundless, limitless, or endless, denoted by the infinity symbol. In modern mathematics it is a concept, and infinite objects such as infinite sets can be studied and manipulated like any other mathematical object.
John Wallis introduced the infinity symbol in 1655. Sometimes called the lemniscate, it has since been used outside mathematics in modern mysticism and literary symbology.
In Zeno of Elea's paradox, Achilles gives a tortoise a head start and appears never to overtake it, because each time he reaches where the tortoise was, it has moved farther ahead. In 1821 Augustin-Louis Cauchy provided a definition of a limit showing Achilles does catch up.
Georg Cantor, near the end of the 19th century, showed that infinite sets can be of various sizes and defined ordinal and cardinal infinite numbers. His theorem proved no largest cardinal exists, since every infinite set has a larger one.
Aristotle, writing around 350 BC, distinguished potential infinity from actual infinity, which he regarded as impossible. Some scholars argue the Hellenistic Greeks had a horror of the infinite, citing Euclid's careful wording around 300 BC, while others dispute that view.
Whether the universe is spatially infinite remains an open question in cosmology. Analysis of cosmic background radiation recorded by the WMAP spacecraft hints at a flat topology consistent with an infinite universe, though a flat universe could still be finite.