Voltage regulation measures how closely the PSU's DC output voltages adhere to the voltages defined by the ATX12V Power Supply Design Guide Specification across the full range of the PSU's output load. A PSU with perfect voltage regulation would have no variation in its output voltages no matter what load is placed on it.
Ripple voltage is an artifact of the AC to DC conversion process by the rectifier. It appears as a high frequency (i.e. choppers switching frequency or a harmonic of that frequency) AC voltage that is superimposed onto the DC output voltages. Capacitors are used to filter out the ripple but its not perfect so some ripple does still get through. Insufficient filtering in the PSU just means that the attached devices own filtering circuits, if they have a filtering circuit, will have to deal with the high levels of ripple. If the ripple is excessive it can cause damage to the capacitors by overworking them on the attached devices (e.g. motherboard, graphics card, HDD's controller board, etc.)