A carbon atom can and does form four bonds, which is among the reasons why orbital hybridization proved to be such a successful theory.
Neutral carbon is located in group 14, period 2 of the periodic table and has an atomic number of 6 and, subsequently, a total of 6 electrons.
Out of these six electrons it has, four are its valence electrons, i. e the electrons located in the outermost shell of the atom. Carbon's electron configuration looks like this
#"C": 1s^(2) 2s^(2) 2p^(2)#
Notice that out of the 4 valence electrons a carbon atom has, only 2 are unpaired and thus available for bonding, the ones located in the #2p_x# and #2p_y# orbitals. So, in theory, carbon should not be able to form four bonds since that would require 4 unpaired electrons, not 2.
Here's where orbital hybridization came into play. According to this theory, when the carbon atom is in an excited state, one of the two electrons located in the #2s# orbital will get promoted to the empty #2p_z# orbital.
Then the #2s# and the three #2p# orbitals will combine to form four #sp^3# hybrid orbitals, like this
As a result, carbon now has 4 unpaired valence electrons with which it can form four bonds.