It depends on where the capacitor is connected.
If placed directly accros the supply wires, then the capacitor plays no direct part in lamp operation, the lamp will work fine without it.
It is there to improve the power factor and reduce the current in the supply cables.
If the capacitor is elswhere in the circuit, then it is probably part of the lamp circuit and the lamp wont work correctly/at all without it.
Some flourescent lamp circuits have a capacitor in series with one lamp, this reduces flicker and improves power factor.
The purpose of a conventional ballast is to limit the lamp current to the required figure by the impedance of the ballast, which is a coil of copper wire on an iron core.
If the lamp voltage is more than about 60% of the supply voltage, then the ballast also incorporated a step up transformer, this is more common in the USA with 120 volt supplies.
Sometimes the transformer has mulitple tappings so as to permit use on different supply voltages.
In some cases a high voltage pulse is needed to start the lamp, the required circuit may be in the ballast, or in seperate starter or ignitor.
The modern tendancy is to use electronic ballasts, these contain an electronic circuit that limits the lamp current, but with less loss or waste than a conventional unit.
