Short answers, pulled from the story.
Poincaré is best known for discovering the first chaotic deterministic system while working on the three-body problem, laying the foundations of modern chaos theory. He is also credited as the creator of algebraic topology, a pioneer of special relativity, the first to propose gravitational waves propagating at the speed of light, and the originator of the Poincaré conjecture, which was solved by Grigori Perelman in 2002-2003.
Poincaré presented the Lorentz transformations in their modern symmetrical form in a paper delivered to the Paris Academy of Sciences on the 5th of June 1905, introduced the relativistic velocity-addition law, and identified the invariant combination of space and time coordinates. He wrote to Hendrik Lorentz in 1905 correcting an error in Lorentz's equations and explaining why the time-dilation factor was correct. Most historians regard his contributions as foundational, though they note that Poincaré and Einstein had different research agendas and physical interpretations.
Poincaré never received the Nobel Prize despite receiving 51 nominations between 1904 and 1912. Of 58 nominations for the 1910 prize alone, 34 named him, submitted by Nobel laureates including Marie Curie, Hendrik Lorentz, and Albert Michelson. Those who nominated him noted that the central obstacle was identifying a single specific discovery, since his contributions spanned too many fields to isolate one.
Poincaré formulated the Poincaré conjecture in the early 20th century as part of his work in topology; it became one of the most famous unsolved problems in mathematics. Grigori Perelman solved it in 2002-2003.
According to the psychologist Édouard Toulouse, who published a study titled Henri Poincaré in 1910, Poincaré worked in two daily sessions: 10 a.m. to noon and 5 p.m. to 7 p.m. He completed entire problems in his head before writing them down, never spent a long time on any single problem, and trusted the subconscious to continue working while he moved to something else. He was ambidextrous, severely nearsighted, and remembered formulas primarily by ear.
Poincaré argued that mathematics is a priori synthetic and cannot be deduced from logic alone, placing him in direct opposition to Bertrand Russell and Gottlob Frege. He held that intuition was central to mathematics and strongly opposed Cantorian set theory. In science, he developed the position known as conventionalism, arguing that principles such as Newton's first law are conventional framework assumptions rather than empirical truths, a view set out in his 1902 book Science and Hypothesis.