Most garbage disposals are sealed using plumber’s putty. You’ll need to detach the garbage disposal, clean off the old putty and reseal. Other types of disposals use only a flange that is tightened and compressed with screws. This may need to be replaced. Either way, this can be a big job if you don’t know what you’re doing (requires removing and replacing drain pipes, as well).
I'm not what you'd call a handyman, although I've been working to get better at that. I will now at least try a home project before passing on it. So, when our 15-year garbage disposal died I had another opportunity. I determined our old unit to be a Badger 5 1/2 HP (this exact item). I didn't have time to wait for the item to arrive from Amazon so I went to Home Depot and coughed up an extra $20 for the Badger 500 - Home Depot's branded version of this item. They are identical except for the name & sticker.
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Update: As an aside, I also learned from my research that the best way to clean my disposal is to use citric acid. Having previously had no idea what citric acid was used for, I decided to order some from Amazon, going with the brand my sister uses: Citric Acid Powder - Ultra Fine Pure Powdered Crystals - Natural Preservative Food Grade Quality (16 oz) and gave it a whirl after 3 months of using my disposal. I let the citric acid sit with some water in the disposal for about ten minutes, and then I let it run for a solid 2 minutes. I repeated the procedure a few times and after about 30 minutes—wow, all the leftover gunk and food was flushed out of the disposal, leaving it clean with a very mild citrus scent. I feel like I’m becoming a real DIY at home kind of guy.
You should first find out whether only the disposal is broken, or if the power has gone out completely in the areas near your kitchen sink. Try resetting the circuit breaker that leads to the kitchen, or replacing a fuse if you have an older electrical box. If the garbage disposal makes no noises at all when you flip the switch, you might also want to check under the sink to see that it’s plugged in.