Why is carbon tetravalent? How does the electron from the 2s orbital "jump" to the 2p, thus leaving 4 unpaired electrons to form four covalent bonds?

Also, does the octet rule not apply to carbon? If it does, how can it bond with only two electrons? It would lack two more.
Look up "orbital hybridization".  It says that atomic orbitals can be combined to make new orbitals for bonding purposes.  Carbon can take 1 s and 3 p orbitals to make 4 sp3 orbitals that can make 4 bonds.