The Good News is that your plumbing looks very good.
Let us begin by helping you learn the proper terms as we go along here.
What you have is a "Disposal Type Compression End Waste Kit" and a P-trap.
Whenever we have two sinks, in the same room, and the center to center horizontal measurement between the drains does not exceed 30" we may combine them on a common P-trap. In order to connect the two sinks we must have a "Waste Kit".
When the "Waste Arm" (pipe coming out of the wall" is midway between the two sinks we would use a "Center Waste Kit" however in your case the Waste Arm is extending out of the wall on the right hand end of the installation so they selected an "End Waste Kit"
On a common waste kit the ends of the cross tubes are turned 90deg upwards so they can attach directly to the tailpiece coming out of the sink but you also have a disposal so they selected a "Disposal Type End Waste Kit". The difference here is that the waste kit cross tube is straight and attached directly to the output of the disposal. On the opposite side we see a TEE which attaches the cross tube to the tailpiece coming down from the second sink. That TEE is actually a specialty TEE called a "BAFFLED TEE" because inside that tee there is a baffle that directs the flow from the cross tube downwards at the Tee. This helps prevent the discharge from that sink from backing up in the second sink. (It is also the most common point to clog up so you should be aware of it for future reference).
The output part of the tee (portion extending downward is called the Tee Tailpiece and it fits directly into the P-Trap input. The inverted U part of the P-trap always traps water in it, which forms a trap to prevent sewer gassess from coming back through the pipe.
All of these pieces are made to "Tubing Standard" and are connected by means of "Compression Fittings" (See Attached illustration)
To assemble a compression fitting we first begin by sliding the compression nut up on the tailpiece of the pipe we want to connect to. We then slide the compression ferrule on the tailpiece. Be very careful when slipping the ferrule on the tubing. The thick flat side goes on first so that the thin pointed end points towards the end of the tube (Pointing away from the compression nut). You now slip the female body section of the new pipe on the existing pipe and push it up to the desired height. Now slide the ferrule down tight agains the body of the new pipe and slide the compression nut down. You then tighten the compression nut. As the nut tightens the compression ferrule is forced further downwards wedging the pointed end between the inner wall of the new pipe and the outer wall of the existing pipe. Continue tightening until you have it "Finger tight" then give it another 1/4 to 1/2 turn to lock it in place. You may need a pair of pliers to finally tighten it but be careful. If you over tighten it you can split the compression nut. If you happen to split the nut you will need to get a new one.
You stated that your pipes are loose and you have had them apart. In order to correct your problem you should loosen the compression nut and slide it up out of your way, then slide the compression ferrule up and wipe the tubing with a rag to clean off any residue that might be present. Then slip the ferrule and nut down, tighten the night as described above and you will be fine.
The only problem I see with your installation is the electrical connection to the disposal. In your installation they ran the Romex cable from your switch directly to the disposal. Properly the Romex should terminate in a receptacle mounted on the back cabinet wall. The disposal should then have a cord attached that has a plug on the end, which would be plugged into that outlet. The purpose of having a plug & outlet is to provide a means of service disconnect to insure that no one could turn the disposal on while you are under the sink working on it.
PS-In the illustration you will see it listed as a DWV compression fitting. The Initials DWV mean Drain, Waste and Vent.