I have disassembled all of the failed compact fluorescent lamps (CFL) I had, from curiosity and in attempt to find out the reason of failure. Of course, I did not attempt to repair them, as I don't understand completely all the details of their construction, although I believe there are cases when the repair (under carefully observed safety precautions) would be possible - when the tube itself (including the filaments) is obviously in good condition and some of the - often underdimensioned - parts of the circuit has failed.
Cracking these beasts open is a little bit more difficult than in other devices I was looking into, as they are not supposed to be opened and are usually sealed using a combination of "clicks" (hooks) in the parts of plastic casing and some sort of glue. Applying certain force usually leads to success, but one should not expect it will happen without any visible damage.
The circuit board is usually stacked between the two pieces of plastic case, one with the tube attached, the other with the threaded contact end. There is not too much space inside, so the board is quite crowded and all parts are mounted in "upright" position. There are two filaments in the tube - one in each end - so there are four wires connected to the circuit board from the tube and of course two more as the power supply. On the picture, the wires are mostly removed, sorry.
The circuit is very well explained on the following excellent webpage: http://www.pavouk.org/hw/lamp/en_index.html. I will just attempt to identify the parts on the circuit board: 230VAC comes in via fuse (B) and choke (M) to the rectifier bridge (A), and is smoothed on capacitor (D). (L) is supposed to be an NTC (thermistor) to limit inrush current, but is probably used only with higher power tubes. Capacitor (K) and diode (J) are part of the startup circuit. The oscillator circuit main constituents are the choke (F), serial capacitor (C), and the transistors of which only is seen (I). (H) is the feedback transformer feeding to the base of transistors. (E) is the capacitor in series with the tube's filaments, which ensures current flowing through - and heating up - the filaments while the arc has not been ignited yet. It has a PTC thermistor (G) connected in series, providing for "smoother" startup.
The CFLs are very good replacement for classic bulbs in places where are used almost continouosly with not many switching on and off. However, the economy and ecology has to be considered carefully, especially when the cheaper types usually don't have the required lifetime, reliability and light output.