Research system
Most nations have between 1 - 3 research centers. For each research center, you select an area of primary and secondary focus. Each turn, every research center achieves a breakthrough. There is a 40% chance of it being in the center's primary focus, a 20% chance of it being in the research center's secondary focus, and a 40% chance of the advance being chosen at random, from among the tech categories you have not yet completed.
Technologies are associated with dull grey stars. Each time you get an advance, a dull grey star becomes a bright yellow star. Once you replace the last dull grey star with a bright yellow star, you unlock the tech. New techs are unlocked at the end of your turn.

In this example, Britain's first research centre has a primary focus of piston aircraft, and a secondary focus of surface ships. Its second research centre has a primary focus of jets and a secondary focus of tanks.
This is Britain's first turn. Note that even before it's done any research, it already has one star towards jet tech, and two stars towards radar. These represent Britain's technological achievements in those areas as of June 1941. Also, jets are special units. (More on that later.) Jets do not replace piston aircraft. Late game, expect to have lots of piston aircraft with some jets mixed in. Piston aircraft research is a good choice for Britain.

Research results are now in. Center 1 got a star towards extended range. Fighters, dive bombers and torpedo bombers start off with a range of 3. But once you get extended range, that increases to 4. Britain's research center 2 got a random tech advance: one star towards industry. Industry tech adds +2 to the unit placement capacity of your factories. The underlying territory must be worth at least 2 to qualify for the bonus.