Simply put, compound exercises are those that work multiple muscle groups at the same time. This is known as muscle recruitment. The more recruitment, the better your fat burning. The more fat you burn, the leaner you get. You will commonly find compound exercises in Cross-Fit and power lifting workouts. However, anyone that really understands their benefits for building strength, muscle and losing fat will use them as the focal point of their fitness program.
The image above shows two people doing pushups with a one-arm row. This directly works the chest, triceps, shoulders and back. Also, as part of maintaining proper form, the muscles in the abdomen are used to stabilize as they change from push up position to the row and back to push up. This is just one example of many possible exercises that you can do to burn more fat in less time. Here are a few of the most well-known.
Bench Press - If you have ever touched a weight or know someone who has, you have at least seen this exercise. You will need a bar, some weight plates and a bench with stands to perform this exercise. Lay flat on your back with your head between the stands. Grab the bar slightly wider than shoulder width and lift it off the stand. Bring it slowly down to your chest while inhaling. Exhale evenly as you lift it back up to where your arms are straight. Repeat for 5-10 reps. Start with just the bar if you are a beginner to practice good form. Then add weight to build muscle. There are variations on body position (incline, decline) and hand position (narrow, wide, overhand grip, underhand grip). As you progress, experiment with these to add variety and confuse your muscles.
Pushups - This is similar to the bench press in effect, without the equipment. Place your hands shoulder width or wider (I like wider to work my chest more) with arms locked. Only your hands and toes touch the ground. Keep your back straight. Lower yourself to within 3 inches of the floor while inhaling. Press yourself back up while exhaling. Beginners can be their knees if they cannot support their full body weight. Do what your fitness level allows.
Squats - Both front and back squats (the bar is either held on your chest or shoulders) work nearly every muscle below the waist plus the lower back and abdominals for stabilization. You will need a stand on which to place the bar so you can walk under it. Set your feet a bit wider than your shoulders for a good base. Grasp the bar and slowly squat down to where your thighs are parallel to the floor. Inhale as you drop. Raise back up to standing in a slow controlled motion while exhaling. This can be done with just body weight for beginners. Do NOT use more weight than you can handle. I also recommend having a spotter for safety.
There are so many more exercises that are compound. Check out this link for more ideas. Many can be done with dumb bells or kettlebells at home. As with any fitness program, do what your fitness level can handle. Doing something is better than doing nothing. Start where you can and improve every day. Persistence and patience are key.