A "successive approximation", such as the standard one in the stm32, gives best results if the analog voltage it is trying to convert is a steady voltage.
This is because the way it works is to generate a voltage with a built-in DAC and sees if this voltage is higher or lower than the "input", updating the DAC value at each comparison. A "binary search" is used so only 10 comparisons are needed to get a 10-bit result.
So how does the ADC ensure that the input voltage is steady?
Simple: It doesn't measure the input voltage. Instead it charges up a capacitor as the first stage of the conversion, and then disconnects that capacitor from the input then measures the voltage on the capacitor.
This 3 - 480 cycles is the time allocated to charging the capacitor.
Once fully charged, it takes a further 12 cycles to get a 12 bit result.
So the charging time might be the dominant thing that determines how fast you can take samples.
Why would you want a long time?
If your analog voltage source is fairly high impedance, it takes a long time to charge the capacitor so it accurately reflects the input voltage.
If the impedance is R, the charge time to 1% (i.e. only 7 bits) is 5 C R, where C is the capacitance used to store the voltage.
ST can't make C too small because then it won't hold the voltage sufficiently well.
And you might have a fairly high R because e.g. if you're measuring battery voltage you use a potential divider, and low resistance values will be a large permanent drain on the battery.
Also note that to be able to turn on and off the charging, ST put in an "analog switch" MOSFET that has a certain resistance r which has to be added to your effective impedance R. So it's never useful to charge for less than 3 cycles even if the input is a solid power-supply-rail.
Hope this helps,
Danish
