Back to basics, Q = CV i.e. amount of charge in a cap = capacitance x voltage across it.
Differentiate to get \$\dfrac{dQ}{dt} = C\dfrac{dV}{dt}\$
Rate of change of charge is current therefore: -
Current in capacitor = \$C\dfrac{dV}{dt}\$
When in parallel with an AC source the current is the differential of the voltage multiplied by capacitance. Presuming that the voltage source is sinusoidal you'll find that current is also sinusoidal and leading the voltage waveform by 90 degrees: -
And for fairness I've shown what it looks like for inductors too: -
When the capacitor is in series with the supply the current depends on how much load resistance is connected to the output of the capacitor and this will produce a current that is somewhere between being in-phase with the supply voltage and leading by 90 degrees. This is dependent on the values of the load resistor and capacitor.
In the top circuit shown by the OP, the capacitor acts as a controlling impedance thus dropping voltage and controlling the current through the LEDs. And, because the voltage across the cap and current through the cap are always 90 degrees apart, the capacitor does not theoretically dissipate power. resistor dropper would generate heat because voltage and current are in-phase.
One of the reasons for an inductor at the output of a regulator is to filter the regulator noise, usually from a switching regulator. The TPS79915 is a linear regulator which should have very low noise to start with, a 3.3uH inductor serves little purpose. Given that, although heavily damped by the load, it may even set up some resonance in the MHz range with the capacitors, which is undesirable.
In the original schematic, there is a FB (ferrite bead). The purpose of that is probably to block high frequency (10's or 100's of MHz) from either going up or down. Without knowing more about system, it is hard to say how much does that do.
In the range of capacitors that you need, I would be inclined to use all ceramic capacitors. Lower ESR and much longer life. The sizes are not critical, 0603 (or 0805) could be reasonable general purpose sizes.
Best Answer
It's not clear what exactly you mean by series versus parallel resistor-capacitor filters. Posting a schematic of each would clarify this.
You probably mean series is something the signal passes thru, and parallel is something that works as a shunt. Note that the same thing that is a low pass filter in series is a high pass filter as a shunt, and vice versa.
Basically a capacitor will block low frequencies and short high ones. If you put it in series with a signal then it is a high pass filter. If you put it accross a signal, it will short the high frequencies thereby making a low pass filter. The value of the capacitance and the resistance it is working against tell you the -3dB rolloff frequency of the filter, whether high pass or low pass. This frequency is:
When R is in Ohms and C in Farads, then f is in Hertz.