Although the series is a highly fictionalised version of the story of the making of the atomic bomb, the basic concepts addressed are the same. The competition between the two teams is based around theoretical concepts of how to make a functioning weapon. In order for a mass of radioactive material to explode, it needs to reach criticality, meaning there's so much of it in one place that the chain reaction which drives the explosion can no longer be controlled. The original, "orthodox" concept on which Akley's team is working is the "uranium gun" model, which involved literally building a cannon inside the bomb casing, which then fired a uranium plug into a larger uranium mass, forcing the total mass to reach criticality. This model was based on the weaker element uranium which meant it needed much more material to work, was not very powerful, and it was the bomb-making equivalent of using a sledge hammer to make it go bang. Winter's team is working on a much more advanced concept: the plutonium implosion bomb needed much less radioactive material, and was based on a design which wrapped the plutonium in high explosive. When the explosive detonated, the force was directed inwards, compressing the plutonium and causing it to reach criticality due to the atoms being squashed closer together. It was far more efficient and far more powerful, and is the design on which modern nuclear weapons are based.
Historically, the two bomb types were produced in tandem because of doubts that they would both work according to their designs. So in the end, the Trinity test used an implosion design, as was Fat Man dropped on Nagasaki. But they also had a uranium gun weapon called Little Boy, and they dropped that on Hiroshima. Trinity's blast yield was the equivalent of about 22 kilotons of TNT, while Little Boy's was 15 kt, and Fat Man's was 21 kt.