Legs do not really need any explanation. Legs are used to help us stand up and move around. The legs are made up of four main areas to be worked out. These areas are: the quadriceps, hamstrings, glutes, and calves. Here is a picture for reference:
The two muscles not displayed are the calves which are on the back side of your shins and the glutes are your butt. For this post, I am going to do one exercise per muscle.
To start, the quadricep help to stabilize the front of the leg and help the leg to run, walk, or just stand. One exercise that can be done to specifically target the quadricep is leg extensions. Like the title of the exercise, leg extensions are done by sitting down (and from a sitting position) pumping your leg forward on the leg press machine. This will specifically target the quadricep.
The hamstrings are the muscle on the back of the hamstring. The hamstring’s purpose is to help bend the knee and tilt the pelvis. The leg curl is a workout that specifically targets the hamstring and also it partially works the calves. The leg curl is done with a machine or a bench that can load weights. The leg curl is done by laying down and bringing your feet towards your butt. A picture showing how to do it:
The glutes or the gluteus maximus is the largest muscle in the body. The glute’s purpose is to help provide stabilization when moving the leg or pelvis, and while it is not its primary purpose, it also provides a nice cushion for whenever we sit. The exercise for this is called hip thrusts. This is done with a barbell and you start with your back leaning against the gym bench and your legs in a crab like position. The barbell should be on top of your lap whenever you thrust, and the end position is with your back nearly flat on the bench with the barbell in the air. The picture for this exercise will explain it better: 
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Finally, calves help us to stabilize our legs and ankles whenever we jump or most leg muscle activities. The exercise for the calves are called calf raises. To do this with a machine, you put a weight you are comfortable with then you put the padding on top of shoulders and go on your tip toes. 
The legs are a very important part of our whole body, and it is essential that people do not skip this muscle group. After all, your legs support you body anytime you walk.
This is a very informative post and the pictures are quite helpful. I’ve never seen a calf raise machine like the one pictured, but it is very interesting. I usually do single-leg calve raises with dumbells in my hand, but the weight on the shoulders seems like it would be more effective.
You do a great job describing the exercises and their purposes, as well as the function of the muscle groups. It is important to work out all areas of the leg. Good job!