Most kitchen sinks contain two basins divided by a partition in the middle. This allows you to wash dishes in a more efficient manner, because one basin is used for washing in soap and water while the second is useful for rinsing and drying. The kitchen faucet swivels from one basin to the other for ease of use so you have access to water for both sides whenever you need it. If the faucet stops swiveling, it may be too tight or it could be stuck on something internally.
Turn off the water supply to the faucet by tightening the valve under the sink. This prevents accidental leakage while you're adjusting the faucet. Turn on the faucet to drain the water still in the system.
Vibrate the faucet back and forth gently to see how much movement the faucet allows. If it moves slightly and stops, it's likely stuck on something. No movement at all indicates it is locked up due to deposits or some other obstruction.
Apply a metal lubricant to the base of the faucet. This loosens up any mineral deposits that may have built up from hard water over time, letting the metal components slide with less effort.
Loosen the faucet attachment with a screwdriver. The specific method varies depending on your sink but you'll likely need to reach under the sink to find the appropriate bolts or screws holding the faucet in position. Other faucets have a bolt at the top of the faucet, often hidden under a cap. Loosening these gets the faucet swiveling again if tightness is the problem.
Disassemble the faucet if it still won't swivel. The means for doing this varies depending on the make and model of the faucet but film yourself taking it apart with a digital camera so you can then reassemble it correctly. Apply lubricant to all the major components and replace any o-rings. If the faucet still won't swivel once it is put back together, you'll likely need a new one.