The absolute magnitude of the eigenvalues (in a linear problem) alone has no meaning at all; it's an artifact of the units you choose to express the problem in.

The chain of comments is getting out of control, so I'm making this an answer. I'm not going to answer the full question; as I said, see wikipedia or the other answers here. I'm just answering the bit that says

Consider two two-dimensional linear ODE systems: first with eigenvalues {-1000000,-0.00000001} and second with {-1000000,-999999}. As for me, both of them are stiff. But if we consider stiffness ratio definition, the second system is not. The main question: why stiffness ratio is considered at all?

Okay, let's consider an example of the second case:

$$y_1'(t) = -1000000 y_1(t)$$ $$y_2'(t) = -999999 y_2(t)$$

Now let's consider a time variable with different units: $t^* = 1000000\cdot t$. Then calculus reveals that

$$y_1'(t^*) = - y_1(t^*)$$ $$y_2'(t^*) = -0.999999 y_2(t^*)$$

Note 1: I chose a diagonal system to make it totally obvious, but if you try it with another system with those eigenvalues, you'll see the same effect, since multiplying a matrix by a constant multiplies its eigenvalues by the same constant.

Note 2: I'm not even discussing here whether the system is stiff. I'm just pointing out that your proposed definition of stiffness (i.e., any problem with $|\lambda|\gg1$) makes no sense, since it would mean that stiffness depends on the units in which I choose to express the problem.

The absolute magnitude of the eigenvalues (in a linear, autonomous problem) alone has no meaning at all; it's an artifact of the units you choose to express the problem in.

The chain of comments is getting out of control, so I'm making this an answer. I'm not going to answer the full question; as I said, see wikipedia or the other answers here. I'm just answering the bit that says

Consider two two-dimensional linear ODE systems: first with eigenvalues {-1000000,-0.00000001} and second with {-1000000,-999999}. As for me, both of them are stiff. But if we consider stiffness ratio definition, the second system is not. The main question: why stiffness ratio is considered at all?

Okay, let's consider an example of the second case:

$$y_1'(t) = -1000000 y_1(t)$$ $$y_2'(t) = -999999 y_2(t)$$

Now let's consider a time variable with different units: $t^* = 1000000\cdot t$. Then calculus reveals that

$$y_1'(t^*) = - y_1(t^*)$$ $$y_2'(t^*) = -0.999999 y_2(t^*)$$

Note 1: I chose a diagonal system to make it totally obvious, but if you try it with another system with those eigenvalues, you'll see the same effect, since multiplying a matrix by a constant multiplies its eigenvalues by the same constant.

Note 2: I'm not even discussing here whether the system is stiff. I'm just pointing out that your proposed definition of stiffness (i.e., any problem with $|\lambda|\gg1$) makes no sense, since it would mean that stiffness depends on the units in which I choose to express the problem.