Mozart wrote during the classical period (Roughly 1720's to 1820's It overlaps with the earlier Baroque and later Romantic periods). He, Haydn and Beethoven were the three most famous composers of this period.
Mozart was a master at combining all the known styles available at that time. He was a master at Italian opera (opera buffa, and opera seria), German Singspiel, polyphony, sonata form and polyphony.
He wrote in every single available genre and produced masterpieces in every genre of the day: Opera, Symphony, Concerto, Sonata, Chamber music, Mass, other choral music. He was arguably most adept at Italian comic opera and piano concertos. (Arguably)
He was supported by the Roman Catholic Church (The Archbishop of Salzburg), and the royal court and worked on commission. He was not a very successful business man, overspent, and made enemies easily, so he was frequently penniless, unlike his father.