A turbocharger is an air pump that is driven by exhaust gas. It pressurizes the intake to allow more air and fuel to go into the cylinder. More air and fuel means more horsepower.
The two sides of the turbo are the turbine side, which is spun by exhaust gases as they leave the engine, and the compressor side, which is driven by the turbine side and which compresses the intake air.
The size of the turbine and compressor wheels, and the size and shape of their housings, affects the efficient range of the turbo.
The turbine size has an impact on how much power the engine can produce. A large turbine will pose little resistance to outgoing exhaust gas, so the engine will be able to make more horsepower, but a large turbine will spin up to speed ("spool up") much more slowly. A small turbine will spool up quickly, but will choke down the exhaust and limit horsepower.
The compressor side has less of an effect on spool-up, but generally a larger compressor will produce cooler compressed air, and will be able to generate more turbo boost, but will take a little longer to spin up to speed.
The moral of the story is there is no free lunch. A larger, more efficient turbo can produce more boost, but will spin up more slowly and will not work as well at low rpms. A smaller turbo will have fast spool-up but weak high-rpm horsepower.
What does this mean? Factory turbos are usually sized for around-town driving, not top-speed autobahn use, so they're smaller turbochargers with low lag and modest maximum boost levels. The 95-99 turbo cars have very small turbochargers, but also exhibit almost no "turbo lag." The 90-94 cars had a bigger turbo, with a bit more lag, but still very fast spool-up.
At the other end of the spectrum are monster race turbos. These turbos will require very high volumes of exhaust gas just to get them spinning. This means no usable boost until 5000 rpms or more. This kind of turbo can produce enormous boost levels and huge horsepower numbers, but the car will be practically undrivable on the street, with weak low-rpm power, abrupt on-boost transition, and huge top-end power. (Have you ever played Gran Turismo and had the Nissan Skyline with all the mods? It's like driving that...)