Sigma bonds form axially, when the two orbitals are parallel to each other (for a primer on orbitals, click here). It is formed when two half-filled orbitals combine and produce a cylindrically symmetric bond about the internuclear axis (or, the axis passing the center of the nucleus). These are stronger than pi bonds and can form between s, p, and even d orbitals.
Pi bonds form perpendicular to each other from the internuclear axis. These, however, are not symmetrical about the axis, and are exclusive to p orbitals (p, pi, makes it easy to remember!). These bonds are the weakest of the two orbital bonds.
For comprehension: an example
Ethylene, or C2H4, forms both pi and sigma bonds. The carbon atoms form a double bond, and in this case, that means it forms one pi bond and one sigma bond. Do not let the image deceive you, there is only one pi bond in this compound. If you look carefully, there is only one electron left over from each carbon atom after the sigma bond is formed in the center. These two join together to form a pi bond, yielding two areas in which electrons can be found (highlighted by the white and green areas).