Let's take ammonia for example (drawn in MarvinSketch below):
The nitrogen is bonded to each hydrogen via an #sp^3# hybridized molecular orbital. What this means is that the #2s# valence atomic orbital of nitrogen increases in energy level until it is degenerate (same energy) with the #2p_x#, #2p_y#, and #2p_z# valence atomic orbitals of nitrogen, and these four orbitals mix together to produce a hybridized molecular orbital.
Since there was one #2s# and three #2p# atomic orbitals, the hybridized molecular orbital is called #sp^3#. The reason why it has to hybridize its orbitals is that it has five valence electrons, two of which are in the #2s# orbital, while there is one valence electron in each #2p# orbital.
To make the three single bonds, nitrogen needs to contribute three electrons (the three hydrogens each contribute one), and so it uses two originally from the #2s# and one originally from one of the #2p# orbitals. The remaining two become the lone electron pair.
