To better understand what turbo is, you should know what normal aspiration is. In a normally aspirated engine, the air fuel mixture or the air (in case of diesel engine) is put into the combustion cylinder via inlet manifold through the downward movement of the piston, the fuel in then burnt to produce energy.
In case of a normally aspirated engine, the amount of fuel that can be burnt in the cylinder is limited by the air or the air-fuel mixture that can be normally sucked into the cylinder by the downward movement of the piston.
To increase the overall efficiency of the engine and increase power output from the same capacity engines, some engines use an exhaust driven compressor to force more air-fuel mixture into the cylinder than the piston itself can suck. So you end up burning more fuel in a small capacity engine. This is known as turbocharging and is one of the methods of forced induction.
A turbo has an exhaust driven turbine at one end and a compressor at the other, the air that's compressed is force fed into the cylinder for the combustion process. In addition to the turbine and compressor, it also has a waste gate that generally regulates the exhaust gas flow into the exhaust side of the turbo and the thereby the amount of boost that is provided.
What advantage does the Turbo offers?
Force feeding more air / air-fuel mixture into the cylinder increases the volumetric efficiency of the engine i.e. you end up making more horsepower from a same capacity engine by burning more fuel within same space. Since they are not driven by engine directly, but rather utilize the waste exhaust gases, it prevents the engine from direct mechanical load.
Why does it lag?
The exhaust gases coming out at low engine speeds do not have enough pressure to overcome the inertia of the turbine / compressor of the turbo. At low engine speed, the low pressure gases are spent to overcome the inertia of the moving parts, therefore, below a certain flow rate, the boost developed is insignificant. The turbo relies on the built pressure of the exhaust gases to create a boost. As the engine speed builds up and the pressure from exhaust gases increase, the turbo starts to spool more quickly developing more intake boost by putting more mixture in the cylinder for the engine to burn. Therefore, in almost every car that utilises an ordinary turbo, you experience a delay between the throttle input and response.
Are there any better types of turbo?
A turbocharger’s performance and size are very closely related. A big turbocharger requires high pressure exhaust gases to spin the turbine, and therefore creating lag at low engine speeds. However, small turbochargers are very efficient and spin quickly at low engine speed, but do not provide the same performance benefits at high engine speeds. The following three turbos are a result of more recent development and are therefore more technically advanced and better.
A twin-scroll turbo uses two different size scrolls that use half the cylinders for each of the individual scrolls. The pressure exhaust wave from an individual cylinder also has a negative pulse that causes interference in a normal turbocharger. To increase the strength of the flow of the exhaust gases, a split exhaust manifold collects the high pressures gases from the complementary cylinders (based on firing order) so that it does not get affected by the back pressure and sends them to the two scrolls, thereby providing a more uniform boost throughout the engine rev range.
Variable geometry turbo
A variable geometry turbocharger is capable of providing boost more uniformly throughout the entire engine operating speeds because of the variable vane geometry turbine. The turbine adjusts itself to the speed and volume of the exhaust gases coming out of the cylinders at different engine speeds. It thereby adjusts the available intake boost as per the varying engine’s RPM making it extremely flexible and responsive throughout the rev range.
Twin-Scroll variable geometry turbo
A variable twin-scroll turbo exploits the benefits of both, the variable geometry and twin scroll turbo. A gate that dynamic adjusts the exhaust flow between the two scrolls provides efficient boost throughout the engine operating range. It uses a simpler method for changing the geometry and therefore does not require more expensive materials as in the case of variable geometry turbo.