Depends on what you are doing with the LED. If it is a simple indicator there is a current limiting resistor that limits the current to around 2mA, and the reverse diode is there to protect the LED from high reverse voltage, as they typically break down at around 5V reverse voltage. If you want less flicker you use 2 LED chips in anti parallel in the same package, so that each one lights alternately. Otherwise you use a bridge rectifier to drive it with pulsed DC.
If you want higher current you will generally use a capacitor in series in place of the resistor, as this reduces the heat produced in the resistor, by virtue of it's reactance. This is used in cheaper lamps along with using a number of chips in series, generally 10-20 LED's in series to get a voltage of around 35V along with a capacitor of around 330nF to do the current limiting, along with a bridge rectifier and very little else.
If you want higher power you will use a dedicated inverter and a transformer to generate the required voltage at around 1A to drive the high power LED units.