Covalent bonds can be classified by how the bonding electron density is arranged relative to the line connecting two nuclei. In this orbital model, an atomic orbital is a quantum state with a characteristic spatial pattern for an electron in an atom; bonding can arise when compatible orbital patterns on different atoms overlap.

A sigma bond, written $\sigma$, has bonding electron density concentrated along the internuclear axis. Every ordinary single bond contains one $\sigma$ bond.

A pi bond, written $\pi$, has bonding electron density above and below—or on opposite sides of—the internuclear axis. It arises from side-by-side overlap of compatible atomic orbitals.

This gives the basic structure of multiple bonds:

• a single bond contains one $\sigma$ bond;
• a double bond contains one $\sigma$ bond and one $\pi$ bond;
• a triple bond contains one $\sigma$ bond and two mutually perpendicular $\pi$ bonds.

The distinction matters mechanically. Rotating one atom around an isolated $\sigma$ bond can preserve the head-on overlap that makes the bond. This is why many single bonds permit substantial internal rotation.

A $\pi$ bond is different. Its side-by-side overlap requires the participating orbitals to remain aligned. Rotating one end of a double bond by about $90^\circ$ would destroy that overlap, so free rotation around a C=C bond is strongly restricted unless the $\pi$ bond is broken.

Carbon bonding geometries

Carbon commonly adopts local bonding arrangements that are described using hybridization as a useful orbital model:

• four electron directions in a tetrahedral arrangement: $sp^3$;
• three directions in a trigonal-planar arrangement, leaving one orbital available for a $\pi$ bond: $sp^2$;
• two directions in a linear arrangement, leaving two orbitals available for two $\pi$ bonds: $sp$.

Thus the carbons of an ordinary alkane C-C bond are commonly described as $sp^3$-hybridized, the carbons of C=C as $sp^2$, and those of C≡C as $sp$.

Hybridization is a model for organizing bonding and geometry, not an additional force or a literal rearrangement that atoms must perform before bonding. Its value is that it connects Lewis bond multiplicity with observed molecular geometry and with the restricted rotation of multiple bonds.

Sigma/pi language will later also help explain why $\pi$ electron density is often more accessible to reagents than the electron density in a $\sigma$ bond.