the smoothing capacitor
A rectifier gives you pulsing DC — a train of humps that still drops to (or near) zero between peaks. Electronics wants a steady level, not humps. The smoothing capacitor, also called the reservoir capacitor, is a big bucket of charge placed across the output that fills in those gaps. It is exactly the local water tank that keeps the tap flowing while the pump is mid-stroke.
Working: each hump charges the capacitor up to near the peak voltage. Then, as the rectified waveform falls away toward zero, the diode stops conducting and the capacitor takes over, dumping its stored charge into the load. By the time it has drooped a bit, the next hump arrives and tops it up. The output is now a high, nearly-flat DC level with a small sawtooth wobble riding on top — that leftover wobble is the ripple.
How much it droops between humps depends on the load and the cap: a heavier load (more current) or a smaller cap droops more. A useful estimate is dV (ripple) is about I times t / C, where t is the gap between humps. Bigger capacitor means smaller ripple but bigger inrush surge at switch-on (the empty cap looks like a short for an instant). This raw, ripply DC is then handed to a regulator to make it truly clean — smoothing is step one, not the finish.
A 1000 uF cap feeding 0.5 A from a full-wave supply (gaps ~10 ms on a 50 Hz line, i.e. 100 Hz humps): ripple is about I times t / C = 0.5 A times 0.01 s / 0.001 F = 5 V. Want 1 V ripple? Use a 5000 uF cap.
Reservoir cap: ripple of about I times t / C; bigger C, smaller ripple.
Bigger is not free. A large reservoir cap charges in brief, tall current spikes only at each peak, stressing the diodes and drawing a big inrush surge at power-on; and the cap must be rated above the peak voltage, not the average.