Stretching a material in one direction usually changes its dimensions in the perpendicular directions as well. Poisson's ratio relates these longitudinal and lateral strains.

For uniaxial loading,

$$\nu=-\frac{\varepsilon_{\mathrm{lateral}}}{\varepsilon_{\mathrm{longitudinal}}}.$$

The minus sign makes $\nu$ positive for ordinary materials that contract laterally when stretched.

Lateral deformation

If a bar is extended longitudinally, its width and thickness usually decrease. Under compression, they usually increase.

Poisson's ratio quantifies how strong this transverse response is relative to the axial deformation.

Incompressible and auxetic behaviour

An ideal incompressible isotropic material approaches $\nu=0.5$ in three-dimensional linear elasticity.

Materials with negative Poisson's ratio are called auxetic: they expand laterally when stretched and contract laterally when compressed.

Poisson's ratio is therefore an independent part of how a material deforms, not another measure of stiffness.