You probably have one of two possible problems.
1) The disposer may be jammed. Check this by making sure the switch is OFF and the breaker is OFF. Insert the proper hex wrench into the bottom hex socket (center bottom of unit) and rotate it left and right to be sure it turns free and easy. If jammed, it may take a bit of forceful back and forth action to clear the jam. Look down inside the disposer with a flashlight to assure there is no obstructions. If all is clear, turn the breaker back on and try the unit.
2) If after doing step one and all is clear but unit still kicks the breaker, the disposer is probably faulty and may need to be replaced.
The answer to your question depends completely on whether this node in the circuit is GFCI-protected. GFCI protection is an absolute must for your disposer; it's a high-amperage electric motor hooked up to your kitchen drain.
Since this J-box doesn't have a standard 3-prong outlet, you'll need to find another outlet on the same circuit; look for a countertop outlet nearby. Plug in an outlet tester (like the one below, available for $5-10 from your local big-box home improvement store), turn on the disposer, and hit the black button on the tester to short hot to ground, inducing a "ground fault".
If the disposer has GFCI protection, it will cut out, and you're golden as far as circuit safety. If not, you definitely need to rewire this J-box with a GFCI outlet.
Even if the disposer is GFCI protected, you have other problems. The outlet is in a "wet" place; you hope that the under-sink area never leaks, but there is always, always, a plumbing emergency at the kitchen sink at some point. To avoid a continuous ground fault through contact with standing water in the J-box (which would prevent you from resetting the GFCI until the whole area dried out), you should seal this area as best you can. A little adhesive spray foam to fill the gap between the wall and the back of the cabinet, followed by a layer of silicone adhesive caulk to waterproof the spray foam (people think spray foam is waterproof, but it really isn't), and a child-resistant outlet and plate with a rubber or neoprene gasket (and/or another dab of silicone) should keep the water out in any situation less than a full flood.
Best Answer
Sounds like the drain after the disposal and connection to the other sink is partially blocked. I'd start with a drain cleaner that's designed for a partial blockage. In the kitchen, it's likely a buildup from cooking oils and grease that got into the drain. If that doesn't work, then you can remove the trap under your sink and remove as much crud from it as you can get. Note that the trap will be full of dirty water, so have a bucket under it when you remove it. If it's still giving you trouble, snake the line after the trap.