Have a look at the trap under your sink. Assuming it's PVC with large plastic nuts, all you should need is ChannelLock pliers:
Unscrew the trap from the wall and disposal, and clean the trap itself, or snake the drain going into the wall, depending on where your clog is. Be sure to have a bucket and towels handy since there will be water backed up before the clog and in the trap.
If you can't find the clog either in the trap or with the snake, and you know it's further down the drain (try running some water through the disposal and into the bucket with the trap out), then I have family that swear by these drain bladders that attach to a garden hose, get inserted into the drain (past the vent) and they swell up to seal your end of the pipe while sending water down until the clog is forced out:
Each one of these will cost around $20 give or take, well less than a visit from a plumber.
I disagree with BMitch as far as if the P-trap will come apart. (Possibly the first time we've disagreed?) The metal nut could not be where it is if the joint did not separate. The rest of his answer I agree is the easiest fix. However, if possible, physically cleaning the trap is preferable to a chemical approach. I'm assuming there is a slip fitting just out of the picture, just as there is in non-disposal side. By detaching the metal ring in the first photo and the unseen slip joint, the trap should be able to be removed.
Even without a slip joint, the pipe could be detached right at the disposal itself. This could actually be preferable as it would allow the P-trap to swivel, which may help in loosening the nut. Are you sure you are trying to turn the nut the correct direction? I'm not saying you don't know how to loosen fasteners. The direction depends on which piece the nut is threaded onto, which is not always clear in plumbing.
The curved shoulder of the nut tells me it is threaded onto the U-bend, so the nut is turned anti-clockwise when looking down at it. It's hard to tell by photos, but you appear to have decent access to the nut from the left side as it is below the pipe coming in from the other side. Even though the opposite trap may be in the way for fully turning a wrench, you seem to have enough room to place the wrench handle near the trap and pull right.
As for counter force, if pulling on the pipe itself isn't working, try a large screwdriver wedged in the inside of the U-bend. If you have the right wrench, the ring nut shouldn't offer a lot of resistance.
Best Answer
There is probably a gasket under that hold down flange. Buy a replacement gasket. Then remove the flange, use sand paper to remove any corrosion until you have a smooth, bright metal surface, then install the new gasket, and screw the flange back in.