Two beds next to each other, a tee shirt, or ceiling rafters in a garage. For some exercises you will need a partner, not a spotter but somebody to do resistance.
Worker and Partner
For simplicity the man performing the exercise will be called the "worker" and the spotter will be referred to as the "partner." Beginning at the top of the body is partner resistance neck rolls. The worker is sitting on a bench while their partner is standing behind them with the palms of their hands pressed against the middle portion of the worker's head.
The worker begins with his chin towards his chest moving his head all the way back. When you are the partner keep in mind not to make it impossible for the worker to do it, but not too easy and have it not work their muscles. Moving down the body, to the triceps you can do dips. You will need two benches, or a bench/bed and a person.
Utilize Your Utilities
Place your palms on the bench/bed with your feet on the other bench/partner. Lower your body as far down as possible and back up working on you triceps. If you are strong enough you can do full body dips using two parallel kitchen counters or two beds together (w/ strong posts, of course).
Place your hands on separate sides of the counter and lower your body down as far as you can go, and push yourself back up. Make sure you do not swing your legs trying to get back up.
Another arm exercise that covers triceps and biceps is the basic push-up. However, there are several variations you can do such as: wide base, narrow base, and diamond push-ups. The first two are self-explanatory, but diamond push-ups are different. You put your hands in close enough to touch, with index fingers extended, and thumbs bent back touching. You perform them by going down trying to place your nose on the ground between your hands.
Biceps and Triceps
My personal pride and joy is a two-person biceps and triceps exercise, which I invented myself. All you need to do this exercise is a tee shirt. One person is on their knees holding the rolled up shirt on the outside. The other person is standing with their hands on the inside. Start with the tee shirt up, and have the bottom partner pull down while the standing partner resists. On the way back up the standing partner pulls up and the kneeling one resists. You are basically simulating a triceps pull-down and a preacher curl.
For working the abs you can do a couple of exercises. First is leg raises, you start in the same position as advanced dips (hands on parallel counters) and raise you legs until they are parallel with the floor, slowly let them down. Make sure you aren't swinging you legs. The second exercise is sit-ups or crunches, which needs no explanation.
In working you quads you can do squats. Since you don't have any weights you need to compromise. If you can't find anything to hold while you do the squats, you will have to have you partner either sit on your back, or if you can't hold him have him press down on your shoulders.
Finally, a killer calf workout is calf raises. They may seem very simple, but you will feel the pain the next day. All you need is stairs about fifteen will be good. If you have less than fifteen you can do as many as you have than return to the bottom and finish your reps. Place your front half of you foot on the step and leave the other part hanging.
One foot at a time, do ten reps in which you go from a resting position to standing on your toes. Then switch feet and go up a step. If done correctly, you should barely be able to walk the next day.
When training without weights you may feel a little awkward doing some of the exercises (having your partner sitting on your back while you do squats) but performing any exercise is better than none. As with any training there is always one simple exercise that takes no equipment, no spotter, and no resistance and has significant impact on your results.
A Slight Difference
This is running. If there is nowhere to run, you can jump rope or any other type of aerobic exercise. As with all training you have to keep up with your diet. You can't slip up and gain five pounds because you ate and did not exercise. Every time you gain five pounds and lose it, it will become harder each time to lose that weight. Keep up with your training and nutrition and you will succeed.