When you apply a steadily increasing axial load to an initially-straight strut after a while it will start to bend, and then, when a critical load $P$ is reached, it will buckle. You could try it with a matchstick.
By considering a general position $x$ on the deformed strut, you can derive Euler's basic formula for $P$ by constructing and solving a second order differential equation.
You need to know two new engineering formulae: The moment due to stiffness is $M = B \kappa$, where $B$ is bending stiffness (a property of the beam) and $\kappa$ is curvature. You also need to know that we can approximate $\kappa$ as $-\frac{d^2v}{dx^2}$, where $v$ is displacement in the direction perpendicular to the initial direction of the beam (see Beam Me Up for a derivation of this).