The TSi is the next generation of the TFSI engines found in the Audi TT and various other VW and Audi models. TFSI (Turbocharged Fuel Stratified Injection). They dropped the "F" to differentiate the new model engine from the old. Essentially they're the same engine with a few changes and improvements.
The word "Stratified" means "To form, arrange, or deposit in layers". With other fuel injection engines, the fuel is injected into the intake ports. With a FSI/TSI/TFSI engine the fuel is injected directly into the cylinder head which creates more torque, more power and uses about 15% less fuel than a conventional fuel injection engine.
The "Stratified" bit relates to the fact that the area around the spark plug is injected with a heavier mixture than the rest of the cylinder head which is injected with a much leaner one. Thus the fuel is injected in different layers if you like. That's why it's faster but more economical.