In a kayak, stability is generally categorized in two ways: primary stability and secondary stability. As a paddler, you want to become familiar with both of these types of stability for your particular kayak.
But there's no such thing as a free lunch...
Primary stability refers to how tippy a kayak feels when you're sitting in the boat in calm water: the flatter the bottom of the hull, the less tippy it will feel.
There is a point at which a kayak will no longer try to right itself and will capsize. This point exists for every vessel, regardless of size or design. The trick is getting to know exactly where that point is for you and your boat.
Taking waves head-on feels more stable because the size a wave would have to be to tip your kayak over end-to-end is pretty extreme. To that end, it's always more comfortable to point your kayak into oncoming waves. But that's not always an option, depending on where you're trying to get to. Taking waves on the beam, or from the side, is different; it doesn't take much to tip a kayak over sideways. How quickly, and in some cases how violently, your kayak reacts to waves on the team is a function of hull shape.
Flat-bottomed hulls have, well, a flat bottom. For these hulls, the capsize point lurks pretty far out on a lean -- but at the same time, the transition from rightside-up to upside-down is sudden and basically irreversible. While flat-bottomed boats feel more stable in calm water (have a high degree of primary stability), they are on the other hand the least stable boats in chop because of the high degree of reactiveness to waves on the beam. In a sense, these hulls have no secondary stability, as the flat bottom is very stable right up to the point where you capsize.
Round-bottomed hulls stem from the traditional kayak hull shape, designed initially for rough seas. These hulls were also intended for use by extremely well-trained paddlers. Because of the round shape, these kayaks tend to rise when encountering waves on the beam, without much reactance. On the other hand, if you tip at all off the vertical, and you don't have strong bracing skills, you are very likely going to capsize.
Depending on the hull shape of your kayak, as you lean further from vertical, you may feel a point at which the kayak doesn't want to tip any further; the kayak will feel like it's bouncing off a barrier in the water. This is called secondary stability.
Chined hulls represent a compromise between flat- and round-bottomed hulls. Generally a V-shaped hull, with a seam, or chine, on the hull where the angle of the V changes, they feel more tippy than flat-bottomed kayaks and about the same as round-hulled kayaks. But as the kayak is tipped further from vertical, the chine reaches the water line and an essentially flatter hull provides more stability; the hull will seem to bounce back from the tipping point. These hulls react far less than flat-bottomed hulls in chop, while retaining a bit of a safety blanket in the secondary stability provided by the chined hull. If you paddle one of these boats, and learn where that secondary point comes into play, you can use both types of stability effectively to paddle in choppy waters.