On Compare.Parts you will find a variety of parts to upgrade your engine's performance.
Engine upgrades really range from mild to wild. For each engine there are different upgrade paths, so we would recommend doing your research on your particular engine as to what upgrades are best suited based on your use of the car.
Breathing mods such as Air filters are a typical starting point with an induction kit fitted to help the engine breath better and often sound better too. The only downside to things like open air filters is the air intake temperatures can rise, negating the performance improvement from increased air flow.
Cooling mods are essential if the car is going to be driven in anger such as a track day, aftermarket radiators by companies such as Mishimoto provide direct replacement for factory/OEM applications making it quite straight forward to install for any car enthusiast.
For big power things like camshafts, pistons, rods and crankshafts are replaced with uprated items. These can affect engine displacement and timing all to release more power from the engine
Turbo upgrades are also really popular and enable engines to increase the amount of force inducted into an engine, this of course will need supporting modifications such as injectors and an ECU re-calibration.