What is meant by “stable/unstable to convection” in the context of the solar atmosphere?

In studying the Schwarzschild stability criterion (for the solar atmosphere); $$\left\lvert\frac{dP}{dr} \right\rvert\lt\frac{\gamma P}{\rho}\left\lvert\frac{d\rho}{dr}\right\rvert$$ From my notes are three quotes that I am trying to understand:

A steep temperature gradient is potentially unstable to convection if the gradient is steeper than the temperature gradient that would be produced by matter rising adiabatically. If this happens then convection will take place.

From the above quote I am getting the sense that for convection to take place there must be an unstable temperature gradient. But what is an unstable temperature gradient?

If the fluid element is denser than its new surroundings it will sink back down, and the material is stable with respect to convection and heat transport will be by radiation. If the fluid element is less dense than its surroundings it will continue to rise, so the material is unstable to convection.

Again, it is the same words that are in bold text which I just cannot understand, this time I would like to know what it means to say that a "material is stable with respect to convection"?

Convection is a very efficient process for transporting heat, with only marginal instability capable of transporting the solar energy output. This means that in convectively unstable regions we can change the Schwarzschild instability criterion from an inequality to an equality $$\left\lvert\frac{dP}{dr} \right\rvert = \frac{\gamma P}{\rho}\left\lvert\frac{d\rho}{dr}\right\rvert.$$

I have no idea what justifies the change of the inequality to an equality (I thought the inequality was required for convection to take place). Now the words "convectively unstable regions" have been used and I have simply no idea what this means.

The questions are basically all regarding the same thing and are summarized in the title. The word convection is constantly being used in conjunction with the words stable/unstable, therefore, until someone please explains what this means in simple English I'm probably never going to understand the Schwarzschild stability criterion in the solar atmosphere.

Basically, this question is not really about the physics taking place here, it is confusion over the use of certain words. If someone would kindly re-phrase "stable with respect to convection" I would be most grateful.