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.