A French press, also called a cafetiere or coffee press, essentially has 3 components: an open-topped cylindrical glass (called a carafe), a signature plunger, and a filter (usually stainless steel). Ironically, the mechanism was first designed in Italy in 1929. Now, it is most popular in Europe and the United States.

Using the French press is fairly simple in case you are new to the manual brewing game. You begin by filling the carafe with coffee grounds. Then, you add hot water and let the mixture steep. Once the steeping process is finished (time frame), simply push down the plunger, which separates the grounds from the coffee, and enjoy your drink.

The pour-over method also requires pouring heated water over coffee grounds; however, with a pour-over, you place the grounds in a cone-shaped component. The grounds are stored at the top of the cone. The grounds are then wet and left to sit for a moment before more water is added. The filter causes the added water to soak through the grounds in a circular motion and eliminates the element of constant contact, so the finished brew is lighter and smoother with no grit regardless of what product you use. You can watch a video on pour-over coffee methods here.

Many people see the lack of grit as the biggest advantage of the pour-over method. If you are looking for a smooth cup that resembles what you could get in a normal drip brew, the pour-over will likely suit you better than the French press.
This method also makes for easier clean-up. With no need for dismantling, this option works well if you are only looking to make one or two cups in the morning.
The flavor of the coffee is less intense when using the pour-over due to the lack of immersion, which is great for people who prefer lighter coffee but is a bit of a bummer for people who gravitate towards strong, full-bodied coffee.