When tuning your car engine you must give attention to other non power related parts of the car to ensure safety and stability.
When you've dramaticallly increased the power of the car you will generally be carrying a higher average speed so your brakes will need to be up to the job of dealing with the extra power.
A brake converts your cars momentum into heat and dissipates this. As speeds increase and braking requirements grow a standard brake disc will start to fail and be unable to dissipate all the heat required.
In bad cases the brake pads will start to smoke, and a layer of gas between the pads and discs will dramatically affect braking ability.
High frictions sports pads will offer a better performance when hot and will resist the tendency to smoke.
Larger discs will be much better at dissipating heat as well so swapping your brakes to a larger set is a sensible upgrade option when performance tuning your engine.
You can also get vented discs where two layers of the heat friction surface are used with a network of channels between them which almost doubles the heat dissipation provided by the disc. These are usually wider than solid discs so you'd need to increase the size of the caliper to carry them and brake effectively.
Most manufacturers offer a higher performance model in each model range and these are generally fitted with better brakes.
It makes sense to source the performance option, specifically chosen for your car model and is generally much cheaper than getting bespoke high performance parts for your car.
Do bigger brake discs needs more pistons and a larger servo?
When adding larger discs and pads you will usually need to increase the number of pistons and size of the brake servo to cope.
The power of your cars braking is limited by the amount of traction the tyres provide so to a big degree you need to match your brakes to the power of the car and the grip.
Often engine tuners will add larger wheels which can accomodate bigger discs and also fit wider tyres to increase the grip and improve braking.
Adding discs with grooves and drilled holes will usually improve the heat dissipation.
High performance sports brake pads will generally not work well in normal traffic as they require some heat to bite. As you'll have them cold most of the time they will be innefective for a second or so before they warm up. They are also often more noisy than fast road or standard brake pads.
Check out online forums such as the one over at TorqueCars to see which brake upgrades will work best for your car and the amount of tuning you have done to your engine.