Without going into too much technical detail, the difference between a traditional mechanical movement and a quartz one lies mainly in the regulating organ, which is what makes a watch measure the time reliably.
In traditional mechanical watches, this is the balance wheel, which oscillates at a rate of 3 to 5 Hz, or oscillations per second. Its mechanism is designed so that each beat of the balance wheel allows the hands connected to the gears of the wheel train to move gradually forward.
On the other hand, quartz watches instead use a quartz crystal. Quartz is an abundant mineral which has some peculiar characteristics.
For example, when crossed by a light electric current, it vibrates at a regular frequency, which is much higher than that of traditional mechanical watches. The frequency traditionally used in quartz mechanisms is 32,768 Hz - orders of magnitude quicker than any mechanical-based regulator - which results in better accuracy of the watch.
In a standard quartz watch, once the crystal has vibrated this number of times, it means that a second has passed. And so, the electronic chip installed onboard the watch orders a tiny magnetic motor to advance the second hand by one step. This "burst" action is one of the most apparent visual features of the quartz movement.
However, there are some modern systems created by Citizen that make the second hand move smoothly across the dial, similarly to what happens with a mechanical watch.