Great question — answering as the UphillTech team:
Sounds like a plugged drain which might be in the outlet of the disposer, but most probably is below where the drains join, maybe in the U-trap. Fill both sinks 1/4 full of water and use a plunger first on the non-disposer sink and see if this clears the blockage. If it doesn't, then use the plunger on the disposer side. Push down easy and pull up vigorously on the plunger to pull the plug out and apart, then see if this clears the blockage. Put a stopper in the other drain when you are plunging on one (prevents the suction action of the plunger from just pulling in water through the other drain). The action you want it to pull on the blockage from below. You may have pieces of wood lodged...
Full answer by Jim Stewart: https://diy.stackexchange.com/a/110363 (CC BY-SA 3.0).
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