With the awareness that there may be some exceptions, we can say that the angle between ideal sp^2 orbitals is 120^@, due to three coplanar electron groups evenly spanning 360^@.
One example of when sp^2 hybridization can occur is when a certain number of electrons need to be donated to other atoms to form a bond, but not all electrons needed are available in the highest-energy orbitals.
Let's say that we wanted to describe the hybridization of one of the carbons in ethene:
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Carbon atom normally has the electron configuration 1s^2 2s^2 2p^2, so it has four valence electrons, but the 2p orbitals are a lot higher in energy than hydrogen's 1s orbital.
To lower the energy of the 2p orbitals so that they can interact with the 1s orbital of hydrogen, they can hybridize with the 2s orbitals to achieve an energy level that is in between those of the pure 2s and 2p energy levels.
This allows access to the 2s electrons.
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Now, we have three degenerate \mathbf(sp^2) hybridized orbitals, formed from one 2s and two 2p atomic orbitals (hence sp^2).
Although the resultant energy is technically closer to that of the pure 2p than the pure 2s, this particular energy level is nevertheless low enough that hybridized bonding becomes favorable, and carbon can now bond with hydrogen with the electrons that originally belonged to the pure 2s orbital. (And carbon can double bond with another carbon to finish its octet.)
We should also notice that there are 3 electron groups surrounding the sp^2 carbon. Additionally, the sp^2 carbon is planar due to the rigid sidelong \mathbf(p) orbital overlap of the two carbons.
Therefore, to distribute the atoms evenly in space, while also keeping the planar structure, the angle should be close to or precisely 120^@.
