A practical definition of the bond classification may be the number of atomic orbital lobes that interact to form the bond. For example sigma bonds are defined by the interaction of one lobe on one atomic orbital with one (and only one) lobe on another atomic orbital. By contrast, pi bonds result from interaction of two atomic orbital lobes on one atomic orbital interacting with two atomic orbital lobes on another atomic orbital. (Just so, delta-bonds
, which you've probably never heard of, are formed by four atomic orbital lobes interact with four atomic orbital lobes, and Phi bonds from 6 to 6). True, pi orbitals are most likely to form between adjacent sets of p-orbitals in a side on fashion, but they may also form between a p orbital and a d orbital, or two d-orbitals. Just like sigma orbitals don't only form between two s-orbitals. This isn't a perfect definition because bonds aren't, strictly speaking, interactions only between pairs of atomic orbitals, but usually a single pair will dominate so using this definition for practical purposes won't lead you astray.
A more formal definition of the bond classification is via electronic symmetry, which is where the sigma/pi/delta designations come from - they have the symmetry analogous to that of their origin atomic orbitals (s, p, d, f). Pi bonds are best defined by the symmetry of electron density distribution with respect to the bond axis, where electron density lies above and below the bond axis, but with a node being on the bond axis.