In a car with a manual gearbox, you can use that stick to your advantage in more ways than the simple act of shifting gears. If you want to slow down, instead of relying on just the brakes, you can pop your gear lever into a lower gear and use resistance from the engine to slow you down. This is called engine braking, and some might think it's harmful to your vehicle. Jason Fenske from Engineering Explained is here to tell you why that's not true.
Letting off of the gas and using the engine to slow the car down is a fine practice. You're closing off the throttle and thus restricting the amount of air your engine is taking in. This creates a vacuum, and that vacuum is what causes the engine to slow down the car as it tries to pull in air. Using engine braking will save you some gas, as well as some minor wear on your brakes.
If you are traveling down a mountain road, engine braking will prevent the brakes from heating up. That can be important because overheated brakes can fade. Instead, you can send the kinetic energy of speed into the engine, where engine braking turns it into heat as the car slows down, and the cooling system takes care of it. When the engine isn't using fuel, it cools off, and the minor heat from engine braking won't have a detrimental effect.
Now, engine braking isn't the most efficient way to stop if you need to do so in a hurry. Your brakes are designed to bring you to a halt more quickly, and they're extremely good at doing just that. If you want to slow down just a bit or slowly, then your engine can help. If you're on a long hill descent, using engine braking to maintain speed is better than using the brakes because you'll eventually overheat them.
Engine braking makes sense in the right situation. It won't hurt anything, and can be safer at times as well. You use the brakes when you need to stop or quickly lose speed.