That annoying humming coming from your fluorescent light fixture originates with the ballast, the interface between the light circuit and the bulb. Your light fixture is probably an older one with an electromagnetic ballast. Stop the humming by replacing the ballast with an electronic one -- assuming you can find one that fits your fixture. You'll also have to replace the older T12 bulbs with thinner, more efficient T8 ones.
What's a Ballast?
A fluorescent bulb is filled with an inert gas -- usually argon -- at a low pressure, and when you pass electricity through the gas, it ionizes and produces light. Ionization creates free electrons, and if you let it proceed unhindered, it would quickly go out of control and overheat the glass. Controlling the current flow is the job of the ballast. Older electromagnetic ballasts accomplish this with magnetic induction transformers; the magnetic force slows down the ionization rate of the gas in the tube. Electricity creates an audible hum as it flows through the coils, though, and the metal plates holding the transformers can also vibrate and make noise. Contemporary ballasts control current flow by means of solid-state circuitry. They are more efficient, and they don't hum.
How Do I Replace the Ballast?
Replacing the ballast is a simple procedure that you can do yourself, provided you can find an electronic ballast that fits your fixture.
Step 1 Access the ballast.
Take the bulbs out of the fixture and set them aside. Remove the ballast cover -- depending on the fixture, you may have to unscrew the cover with a screwdriver or simply unsnap the clips holding it and pull it off.
Step 2 Disconnect and remove the ballast.
Snip all the wires connected to the ballast with wire snippers. Cut them a few inches away from the ballast. Locate the screws holding the ballast to the fixture, unscrew them and let the ballast fall into your hand.
Step 3 Find a replacement.
Take the ballast to a lighting supplier to find an electronic ballast with the same shape. It doesn't have to be identical, but it must be able to fit inside the fixture, and the screw holes must align with those on the fixture. Buy new T8 bulbs when you buy the new ballast -- your old T12 bulbs won't work with it.
Step 4 Replace the ballast.
Screw the new ballast into the light fixture, connect the wires to the house circuitry according to the directions with the ballast and cap the wire connections with wire caps. Replace the ballast cover and install the new bulbs.