Initial (or primary) stability is how hard the craft resists being tipped from the upright position. If you are looking at a graph of stability (see the XL review's stability graphs or any recent issue of Sea Kayaker magazine) the steepness of the angle off of zero is an indication of the primary stability.
Secondary stability is a lot harder to define. Most experienced kayakers will tell you they know it when they feel it (sort of a seat-of-the-pants thing).

One designer used to claim there is no such thing as secondary stability and any kayak that is more stable initially will be more stable at all angles of lean. I'm not sure he still claims this because I once showed him some stability graphs (Chinook and Puffin in the Winter 1986 issue of Sea Kayaker magazine) where the less initially stable kayak, the Puffin, had not only higher relative stability at higher angles of lean than the Chinook but also a higher maximum stability and a greater total area under the curve). Personally, I define secondary stability subjectively as how secure you feel when you have leaned the kayak well to one side.

stability and the paddler Generally the person in a rec boat is a passenger only, yes he/she provides the power to move the boat, but the boat provides the stability. As you get into skinnier boats the paddler takes on more and more responsibility for the stability as the boats get narrower. After awhile some paddlers may prefer boats with what is normally called low primary stability as this generally means the boat is more performance oriented, more suited to the more advanced paddler’s skills. Personally I get bored very quickly now in a kayak wider than about 21", I want to be more involved. I do very well though remember not feeling this way not that many years ago.

Its all relative A couple of canoe examples - my first solo canoe was a Yellowstone Solo. It felt tippy at first, but if I’m lucky, I can now keep it upright most of the time.

When I started doing whitewater, I bought a Dagger Encore – it’s a big boat by today’s standards, but good paddlers can lean it right down to the gunwales with no problem.

This past weekend I had an opportunity to paddle an Esquif Prelude – tiny little boat that twitches like crazy in flatwater, but firms up nicely with a little current.

secondary stability is illusionary I was confused by initial and secondary stability, and the reason is that initial is definable, and secondary is illusionary.

Confused?

As said earlier, initial can be realized and felt as resistance to tipping over. If you overcome the resistance, the craft flips out from under you.

Secondary is quite different, in that there is no resistance, or very little, as you continue the lean. The idea is that you will correct the tippy lean on your own to return to upright before you go over. You can therefore lean to even let water come in, and then correct the lean to return upright. Therein is the incredible secondary stability that is advantageous for leaning maneuvers. However, if you are expecting some resistance to kick in somewhere from secondary stability to prevent a capsize when the gunnel goes into the water, it will not be realized. If you continue the lean, you will capsize. So, secondary stability is really not the same as initial stability in terms of helping you to keep upright.

Initial stability assists in keeping you upright, up to its limitations. Secondary stability enables you to return to upright, if you can do it on your own.

Think of it this way. Forget a paddler or sack of potatoes - just a boat.

Two boxes, both are 5 ft long and have sides three feet high. One is 1 foot wide the other is 3 feet wide. The 1 ft wide one will topple over easier in any waves than the 3 ft wide one and therefore has less initial stability. The ratio of it's sides to the bottom are greater.

If you push your finger against the two boxes and gently push them over, there will be an initial resistance (initial stability) and then, the thinner one will topple over quickly (no secondary). The wider one will not give up as easily as you get to the point of no return. Under water a bit of the side and bottom form a "V" shape which is almost like a "V" shape hull until the sides height / weight over-balance that resistance. (secondary).

All boat building deals with this in the decisions it makes with the height and weight of the above-water-structure and load carrying capacity. In kayaking, usually the ratio of secondary to initial stability run the same. Low initial stability boats never have very high secondary. If that was so, the graphs in Sea Kayaker test would show an upward travel of the line (primary) and then the line would travel away and still upward - and that never happens. Once the initial height of the curve is reached, the line always curves downward. The further the line travels before it goes to the base line of the graph, the greater the secondary.

But one can design boats that provide more, or less, resistance to finally tipping over. And some designers apparently put some thought into this. What you are saying is pretty much correct when applied to simpler hull contours, but it is not absolutely true. It depends on portions of the sides of the hull that are not normally under water. Flared versus tumblehome sides are the simplest examples. You can have the same hull bottom, sitting in the water and providing what one calls “primary” stability, and then you can deside what to do with the sides above the resting waterline.

Not entirely so Some hulls firm up admirably when heeled to the rail or even rail underwater (canoe). The shouldered sides of the Bell boats are one example…Other boats have that high shoulder that is essentially a flare cut off below the gunwale to allow for a tuck.

Some boats have no flare and have their max width further down. Those are the ones that have an initial increase in stability followed by a significant and sometimes sudden decrease in stability when heeled. Many Wenonahs fall into this category( are intended for the upright sit and switch crowd). To their credit its tough to get a Wenonah to start to tip though.

In a canoe you should be able to ride out anything by having loose hips and lips inside the gunwales of the canoe provided the sea does not slop in over the side.

Don’t underestimate the importance of primary stability for feeling comfortably safe in a boat.

It may help to think of stability as a curve, where each degree of heel has an amount of righting force. This pdf file provides a good overview of stability:

Plenty of resistence in some boats A big person can lean a boat right over and not realize they did it, granted. But there is a quite discernible hardening up in my kayaks as well as my canoe when they go over on edge. Enough to give me head room before I capsize, for sure.

you can use actual numbers designating how much force it takes to heel a craft over from level, you put X lbs on the sheer and the kayak tips. If it tips a little it's stable, if it tips a lot it's unstable.

There's nothing "stable" about a boat near it's capsize angle because it's about to tip over without a dynamic response from the paddler. The term "rotationally responsive" is a better description of what happens near the angle of capsize than "secondary stability". The kayak isn't "stable" near the capsize angle, it's about to tip over but if the forces allowing it to tip over aren't that great the forces required to bring it level aren't that great.

What makes more sense is to have a range that you can control the kayak through the arc above and below the capsize angle, that requires a paddler in the kayak, the kayak doesn't do it itself and it varies according to the height/weight of the paddler.