Electrons behave like waves

Electrons have wave "like" properties and behave similarly to sound or light waves.  There are two main theories of bonding, Valence bond and Molecular Orbital Theory.  Moleculer orbital theory relies on the wave properties of electrons.

Recall from introductory physics that a wave oscillates about the axis going from regions of positive phase (red - above axis) to negative phase (blue - below axis) etc.  

The form or shape of the wave depends on the type of orbital the electron is in.  For example, an electron in an s orbital would look as follows in which there is one phase.

On the other hand, p orbitals have both positive and negative phases at the same time.  

Bonding is Constructive Interference

Bonding can be thought of as constructive interference between two waves.  Imagine bringing two separate hydrogen atoms together (H₁ and H₂ below) each with one electron in its 1s orbital.  If the phase of the two electrons is the same then we can have constructive interference (i.e. a bond).  Constructive interference results in strengthening the interaction between the two atoms and an increased probability of finding the electrons between the two atoms.

The resulting combined wave would have a greater amplitude between the two hydrogen atoms.

Destructive inteference Results in No Bonding (Anti-bonding)

In contrast, if two waves are brought together but one of them has the opposite phase of the other we have the possibility of destructive interference.

Note how the two waves cancel a portion of the other forming what looks like two separated waves.  This does not form two waves but rather one wave with a region of zero electron probability in between.  With this situation, no bond would form.