Turbochargers and superchargers both work by blowing air into the engine, rather than just letting the engine suck air in.
Superchargers are driven by the engine, are instantaneous but take power and fuel to drive them.
Turbo chargers are driven by "free" exhaust gases but require the engine to be producing enough to drive the turbo, hence the delay.
When you accelerate gently you don't use the turbo (hence no lag); if you accelerate hard the electronics "ask" for the turbo and the engine has to wait for the exhaust gases to spin it up. Some cars exhibit the delay more than others. My first Volvo V70 2.5D had no noticeable turbo lag; my current Volvo 2.4D had noticeable lag until I had the engine re-chipped. This can be expensive (a few hundred pounds) and you have to declare it to insurance companies. I have never had premium added or cover refused for this but some companies/brokers need you to speak to them and will not do an online quote.