Euler's buckling theory describes the critical load at which slender columns fail due to buckling. It is based on the relationship between the column's length, cross-sectional area, and material properties. When a column is subjected to axial compression, it can become unstable and bend, leading to failure even if the material is not stressed beyond its yield strength. The theory provides a formula to calculate the critical load, which depends on the column's moment of inertia and its effective length. This helps engineers design safe structures by ensuring that columns can support the expected loads without buckling.