Short answers, pulled from the story.
Mathematics includes number theory, the study of integers and their properties; algebra, the study of operations and the structures they form; geometry, the study of shapes and the spaces containing them; analysis, the study of approximating continuous change; and set theory, which is currently used as a foundation for all of mathematics. The contemporary Mathematics Subject Classification lists more than sixty first-level areas.
In Latin and English until around 1700, the term mathematics more commonly meant astrology, or sometimes astronomy, rather than the modern discipline. The meaning shifted gradually between roughly 1500 and 1800. This caused mistranslations; for example, Saint Augustine's warning that Christians should beware the mathematici, meaning astrologers, is sometimes wrongly read as a condemnation of mathematicians.
At the end of the 19th century, mathematicians realized the intuitive definitions of basic objects were not rigorous enough, exposed by non-Euclidean geometries and Russell's paradox, and inflamed by Cantor's discovery of different sizes of infinity. The crisis was resolved by systematizing the axiomatic method inside a formalized set theory, an approach embodied in David Hilbert's formalism around 1910.
Fermat's Last Theorem, stated in 1637, was proven in 1994 by Andrew Wiles using tools including scheme theory, category theory, and homological algebra. Goldbach's conjecture, stated in 1742, which says every even integer greater than 2 is the sum of two primes, remains unproven despite considerable effort.
The Fields Medal is considered the most prestigious award and the mathematical equivalent of the Nobel Prize. It was established by the Canadian John Charles Fields in 1936 and is awarded every four years to up to four individuals. Other major honors include the Abel Prize, first awarded in 2003, and the Wolf Prize in Mathematics, instituted in 1978.