In the simplest sense, the bond order is equal to one-half the number of shared electrons between two atoms.
For example, in acetylene, which can be written as
the bond order between each H and C is 1 because two electrons are shared (1 from H and 1 from C).
However, there are 6 electrons shared between the two C atoms (3 from each C atom), so this bond order is 3. This simple picture covers most stable molecules.
However, with radicals and ions, a more complicated approach is required.
When considering the molecular orbital diagram, the bond order is equal to
For example, for