A turbo charger is where exhaust gases are used to spin a turbine which is used to compress the air intake. The reason for turbo lag is because it takes a bit of time for the exhaust gases to turn this turbine.
A super charger is attatched directly to the engine to spin the compressor. Thus no lag. But is more expensive and is used mainly by Mercedes. Kompressor.
Increase compression ratios - This is effective up to a point when the mixture will simply spontaneously combust. To prevent this F1 cars use high octane fuels which are more stable and less likely to spontaneously combust.
Compress air intake - Done using turbo chargers or super shargers. However if you pressurise too much then it will damage the pistons of the engine and you will be looking at an engine out rebore!
Cooling - Compressing air raises its temperature! However you want the cold air because the hotter the air is, the less it will expand when combustion takes place. Therefore the turbocharged cars usually have intercoolers which cools the compressed air before it reaches the cylinders.