The act of converting AC into DC is called rectification.
There are two basic types of rectification: full wave, and half wave.
Full wave rectification takes 4 diodes and flips the lower half of an AC waveform upside down and superimposes it over the upper half:
Half wave rectification only uses 1 diode, and effectively cuts off the lower half of the waveform:
The Arduino has a single diode in-line with the power input jack to protect the system against reverse polarity connections. This diode acts like a half wave rectifier.
A capacitor after a rectifier smooths off the peaks and troughs of the waveform:
If the ripple voltage remains above the minimum level of the voltage regulator, then all will be well. The ripple voltage, however, is a function of the current drawn - the more current you draw the faster the capacitor will discharge and the more ripple you will get.
You will be fine running with an AC adaptor at low currents, but there will be a cut-off level where you will start to get glitches and voltage drops on the 5V line when you draw too much current.