Two things can make your garbage disposal hum when you turn it on. The first is a clog or jammed object in the gap between the grinding plate (where the grinding blades are located) and the side of the canister. In most cases, that's easy to fix. The second is a burned-out motor. Servicing a burned-out motor is a job for a pro, but replacing a disposal is a reasonable DIY project; compare costs before deciding to pay for repairs.
Unsticking a Clogged or Jammed Disposal
Many garbage disposals have a socket fitting in the bottom of the unit that allows you to turn the grinding plate manually, using a special wrenchette, which comes with the disposal. If you don't have the wrenchette, you can use a 1/4-inch Allen wrench instead. If your disposal doesn't have a socket fitting, you'll need a wooden implement, such as a broom or hammer handle, to move the grinding plate from inside the disposer chamber.
Step 1: Disconnect the power.
Turn off the garbage disposal switch, then -- as insurance -- unplug the disposer from the wall receptacle, or turn off the breaker in the main panel that controls the garbage disposal.
Step 2: Loosen the clog.
Rotate the grinding plate back and forth to free the clog: If your unit has a socket fitting, insert the wrenchette into the fitting and use it to move the rotor. If your unit doesn't have a fitting, insert a long wooden implement, such as a broom handle, through the mouth of the disposal unit, wedge it against one of the impeller blades and use it to move the grinding plate back and forth.
Step 3: Retrieve debris.
Use a pair of tongs -- not your hand -- to retrieve debris that comes loose. Hard materials, such as olive pits or small pieces of gravel (from watering plants in the sink), can get stuck between the grinding plate and disposal canister; fish these out with the tongs, if applicable. Move the grinding plate again and retrieve any other debris that comes loose. Continue in this way until the plate spins freely.
Step 4: Test the unit.
Turn on the breaker or plug in the disposal, then run water from the kitchen faucet and turn on the disposal. If it still hums, turn off the power and repeat the procedure for clearing the clog.
Garbage disposal units have an internal breaker that cuts power in the event of a clog. If your unit is humming, it means that the breaker hasn't tripped, and there's no need to press the red reset button. If you get no response when you turn on the disposal -- not even a hum -- you do have to press this button. It's on the outside of the canister, either on the bottom or on the side, near the bottom.