I remember when I actually did a whole load of research into these programmes when I was 17 and was thinking about a career in aviation. Of course, I didn't have the money for any private flight lessons to get a PPL or anything like that, so I was more or less limited to using a flight simulator at home.
Anyway, basically, you will spend about 2 years in the airline's own flight school, and you will learn to fly a variety of small single engine Cessnas, Grobs and what not after the first bit of theory. Once you're competent at flying small piston engines, they'll stick you into a twin engine piston engine aircraft and teach you to fly those. Some airlines may also have a jet like a Cessna Citation and you'll learn to fly a jet before they will give you a CPL which allows you to be a first officer. You'll also have a provisional ATPL which will become a full ATPL once you complete the required flight hours with the airline (which can be during revenue flights). After that, you'll be type rated in the Airbus A320 family. After successfully being type rated in the A320 family, you will then go on to work for the airline for at least 7 years, paying off the €40,000 cost it took to train you. Once you have your ATPL and seniority in the airline, you may apply to "upgrade" to another type rating which the airline has.
Within Europe, British Airways, Aer Lingus and Lufthansa all do the same thing. Do note, that with Lufthansa, they may not always give you a job in Lufthansa, but instead in a Lufthansa group airline like Eurowings, Lufthansa Cargo, Austrian Airlines, Brussels Airlines and LGW (the company which took over the Air Berlin fleet).
Note that many airlines with cadet programmes will have you spending a lot of time in the US, where you will be flying small single engine piston aircraft for several hundred hours in places like Arizona. The clear weather allows them to fly for as many hours as possible.