Short answers, pulled from the story.
Teller earned the label because he championed fusion weapon research from the early 1940s and co-developed the Teller-Ulam design with Stanislaw Ulam, the classified 1951 breakthrough that made a practical megaton-range hydrogen bomb possible. After the first thermonuclear weapon using that design was detonated on the 1st of November 1952, the press attributed the weapon to Teller and the title stuck.
The Teller-Ulam design is the classified configuration for thermonuclear weapons, apparently involving separation of fission and fusion components and use of X-rays from the fission stage to compress and ignite the fusion fuel. Credit for the invention is disputed: Teller said in a 1999 Scientific American interview that Ulam did not contribute the key idea, while Ulam maintained Teller produced only a more generalized version of his original design. Hans Bethe described Teller's contribution as "a stroke of genius" in 1954.
Teller testified that he believed J. Robert Oppenheimer was loyal to the United States but that he had seen Oppenheimer act in ways that were "exceedingly hard to understand", and that he would feel more secure if public matters rested in other hands. He was the only member of the scientific community to testify against granting Oppenheimer clearance, and the testimony led to lasting ostracism from much of the physics community.
Teller warned about climate change in December 1957 in an address to the American Chemical Society, stating that fossil fuel burning since the mid-nineteenth century was raising atmospheric carbon dioxide concentrations and would raise surface temperatures in the same way a greenhouse does. He repeated the warning in 1959 at a centennial symposium for the American oil industry, stating that a ten percent increase in carbon dioxide would be sufficient to melt the polar ice cap and submerge New York.
Project Chariot was a plan to use a multi-megaton hydrogen bomb to excavate a deep-water harbor more than a mile long and half a mile wide near Point Hope, Alaska. The Atomic Energy Commission accepted Teller's proposal in 1958. The project was abandoned in 1962 after critics raised concerns about radiation risks to local wildlife and the Inupiat people near the site, and the harbor was found to be ice-bound for nine months of the year.
Teller predicted the Jahn-Teller effect in 1937, which describes how certain molecules distort and affects the chemical reactions of metals. He co-developed the Brunauer-Emmett-Teller isotherm with Stephen Brunauer and Paul Emmett, a mainstay of surface physics and chemistry. In 1953, with Nicholas Metropolis and others, he co-authored a foundational paper for the Monte Carlo method in statistical mechanics. He also contributed to Thomas-Fermi theory, a precursor of density functional theory, and to Gamow-Teller transitions in beta decay.