An example of carbon with #sp^2# hybridized atomic orbital is alkene, specifically the two carbons involved in the C=C. Each of that carbon has 3 sigma bonds and 1 pi bond.
Carbon is tetravalent (forms 4 bond) and the ground state electron configuration cannot explain its valency since there's only 2 unpaired electron (left image below).
Therefore, one of the electron in 2s will be promoted to the empty 2pz orbital (middle image below).
In order to form the 3 sigma bonds in ethene, one 2s and two of the 2p orbitals will mix to form three #2sp^2# hybridized orbitals (right image below). These three hybridized orbital will bond with other atoms to form sigma bonds. The remaining one 2p orbital will form a pi bond with 2p orbital of the other carbon through sideway overlap.
Note that the three #2sp^2# hybridized orbitals are all degenerate and have lower energy compared to 2p orbital. They will arrange themselves into trigonal planar geometry. Also, they have 33% s character [1/3*100 = 33%].