This article has a very clear picture to show how the fluorescent lights with starter work (and confirms Francesc Casanellas' explanation):

http://home.howstuffworks.com/fluorescent-lamp4.htm

In short, when you have a choke and starter, the starter is in the current path only as long as the fluorescent tube is not conducting (gas not ionized yet). Then after the gas is ionized, the fluorescent tube shunts the current away from the starter.

I don't know if you are familiar with the old Kettering ignition system in cars (also called point ignition). During the starting phase, the starter in these fluorescent fixtures behaves very much like the points in the Kettering ignition. Current flows through the starter at first, and current flows through the choke (as current flows through the coil and the points in the car, while the points are closed). This current flow through the starter causes the starter to open soon, as a bimetallic strip heats up in the starter (much like the points open in the car after the distributor shaft rotates a little). When the starter is opened suddenly, just like when the points in the car's ignition are suddenly opened, the current flowing through the choke/coil cannot stop suddenly. So instead, voltage builds up until the current can force its way across the fluorescent tube, or the spark plug gap in the car. (There's a little extra detail in a car's coil, in which the spark plugs connect to a higher voltage secondary coil.)

In the fluorescent tube, if the gas in the tube does not ionize at the first attempt, then current will flow again through the starter, repeating the cycle.

Choke is a inductive coil(wire wound on iron core, made up of iron stampings). When current flow from choke, magnetic field will developed, strength of which depends on inductance(L) in Henry, or L= do/di (variation of flux(do) to the variation of current(di). Voltage induced in choke, could be given by equation, V = do/dt (variation of flux with variation of time), or V= L(di/dt). To get  di/dt starter is provided in series with choke, or starter, brings the current from value i to zero in small time(switch off time) dt, and voltage V developed across choke, sufficient  to produce spark and continue the flow of electrons in fluorescent tube to glow, hence once the tube light glow, no need of starter, you can use it, to glow another tube light, what electricians are doing in function decoration used tube lights (starts many using one starter).

To ionize the gas you need a high voltage and a high electrode temperature to emit electrons. The starter connects the heatting filaments, so they start to emit electrons and after a short time  opens the circuit. The current flowing through the choke continues to flow, because the energy in its iductance has to go somewhere, charging the parasitic capacitances and the voltage increases until the gas ionizes. Then the voltage in the tube drops (typicaly to about 100 V), the current keeps the electrodes hot and, because the ionized gas has a negative resistance, the impedance of the choke limits the current.

If the lamp lights, because the voltage drops, the starter has not enough voltage to start again, but if it happens that the start operation fails, i.e. because the opening of the circuit occurred when current was low (we deal with a.c.) the starter will repeat the operation.

This article has a very clear picture to show how the fluorescent lights with starter work (and confirms Francesc Casanellas' explanation):

http://home.howstuffworks.com/fluorescent-lamp4.htm

In short, when you have a choke and starter, the starter is in the current path only as long as the fluorescent tube is not conducting (gas not ionized yet). Then after the gas is ionized, the fluorescent tube shunts the current away from the starter.

I don't know if you are familiar with the old Kettering ignition system in cars (also called point ignition). During the starting phase, the starter in these fluorescent fixtures behaves very much like the points in the Kettering ignition. Current flows through the starter at first, and current flows through the choke (as current flows through the coil and the points in the car, while the points are closed). This current flow through the starter causes the starter to open soon, as a bimetallic strip heats up in the starter (much like the points open in the car after the distributor shaft rotates a little). When the starter is opened suddenly, just like when the points in the car's ignition are suddenly opened, the current flowing through the choke/coil cannot stop suddenly. So instead, voltage builds up until the current can force its way across the fluorescent tube, or the spark plug gap in the car. (There's a little extra detail in a car's coil, in which the spark plugs connect to a higher voltage secondary coil.)

In the fluorescent tube, if the gas in the tube does not ionize at the first attempt, then current will flow again through the starter, repeating the cycle.

The choke connected in series with the fluorescent lamps serves two purposes.The answers above explain the first - production of the high voltage 'kick' needed to help ionize the gas in the tube.

A fluorescent lamp does not require the full supply voltage once it is turned on. For example, a conventional 4-ft tube requires about 80-100 volts across itself to produce the specified lumens. The choke supports the rest of the supply voltage.

The 'choke' or magnetic 'ballast' is a coil with a magnetic core (using silicon steel) that performs three functions.

The fluorescent lamp has two electrodes at its two ends but the voltage needed for breakdown is very high, much higher than the supply voltage. By providing heating at each electrode, thermionic emission is generated (in a 'hot cathode' type of fluorescent lamp), which helps to ionize the gaseous medium and help reduce the breakdown voltage. Still it is higher than the supply voltage. Thus, the supply lines are connected to the two ends of the lamp (one end of each filament) via a series 'choke' while each of the other end of the filaments are connected across a 'glow' type starter. The starter has a gas with two electrodes, designed to strike a glow above 100V.

When the circuit is energized, since the fluorescent lamp is unable to strike at the supply voltage, the supply voltage is applied across the 'glow' starter, which strikes a glow. Current flows through the closed path, causing heating of the filaments and resulting thermionic emission. During this interval, the current is limited by the series filament resistances and the 'choke'.

Due to the resulting small heating, a bi-metal strip inside the starter bends so as to short circuit the electrodes. At this time the starter is by-passed and the circuit current is at its highest, limited by the 'choke'. Since there is no glow now, the bi-metal strip cools and thus bends back suddenly (as it is designed to do so), interrupting the circuit abruptly, causing the energy stored in the inductance of the series 'choke' to create a high voltage kick, that is now sufficient to strike the heated fluorescent lamp.

Once the lamp is on. the voltage across it drops below about 100V, Thus even though the starter remains connected across the lamp, it cannot strike a glow. During this period, the lamp current is limited by the 'choke' which also drops the balance of the voltage between the supply and that of the lamp arc voltage.

In 'electronic' ballasts, the filament heating, high voltage kick generation and current limiting are all done using a self-oscillating high frequency resonant inverter circuit, which operates from the rectified ac supply.

Ihttps://www.youtube.com/watch?v=z55566ep0Hg