Retrofit LED lamps for light bulbs in E27 and E14 screw sockets and for halogen bulbs are easily available, even at food discount stores. Lately, also LED tubes as replacement for fluorescent tubes are available, sometimes at suprisingly low prices.
How do they work? First, how do fluorescent tubes work? The tube has four contacts, two on both ends, between each there is a filament. In the figure below the full circuit is shown: There is a series connection of the inductor as ballast, the right filament, the starter, and the left filament.
When switched on, the full 230V AC are across the starter. "[...] There will be a glow discharge across the electrodes in the starter lamp. This heats the gas in the starter and causes one of the bi-metallic contacts to bend towards the other. When the contacts touch, the two filaments of the fluorescent lamp and the ballast will effectively be switched in series to the supply voltage. The current through the filaments causes them to heat up and emit electrons into the tube gas by thermionic emission.
In the starter, the touching contacts short out the voltage sustaining the glow discharge, extinguishing it so the gas cools down and no longer heats the bi-metallic switch, which opens within a second or two. The current through the filaments and the inductive ballast is abruptly interrupted, leaving the full line voltage applied between the filaments at the ends of the tube and generating an inductive kick which provides the high voltage needed to start the lamp."
The current flows through the inductor and through the ionized gas inside the tube. The filaments are now used as mere electrodes. Due to the voltage drop across the inductor, the starter (which is in parallel to the lamp) sees a lower voltage (approx. 110-150V) which is below its ignition voltage.
"The lamp will fail to strike if the filaments are not hot enough, in which case the cycle repeats; several cycles are usually needed, which causes flickering [...]" (quoted from http://en.wikipedia.org/wiki/Fluorescent_lamp).
For the LED tube you have to replace the starter with the supplied replacement starter, which is a simple short. One end of the LED tube itself is also a short. Therefore the full 230V AC are seen on the other end of the LED tube. This is where the internal power supply is located. This is smart, because you can install the LED tube in either way, it will always work.
Note that with this simple replacement starter, the inductor is still in series, which causes ohmic losses. Most LED tubes also allow the direct application of 230V AC to the power supply pins. Be sure to measure the resistance on both ends to tell the short from the power supply apart. The short should be near 0 Ω (I measured 0,3 Ω including contact resistances), the power supply of my tube varies strongly around 900 kΩ, and the resistance between both ends is too high to measure. If you modify a whole luminaire, be careful not to insert the LED tube with swapped sides, because this will short circuit the 230V line voltage.