There are two kinds of light meters. They either measure incident light, i.e. the amount of illumination that actually falls onto the subject or reflected light, i.e. the amount of light reflected into the camera from the subject.
You can recognise incident light meters by the translucent half-dome at the top. Underneath the half-dome is a light receptor. By its very nature, the half-dome functions as a tool that averages the light coming from all directions. When you use an incident light meter, you need to go to your subject and hold the light meter towards your camera. (There are more advanced uses for these meters: you can also measure the relative intensity of light hitting your subject from various directions, as well as getting a simple and fast reading for the amount of ambient light.)
Light meters measuring reflected light are more common these days. Every single camera with a built-in light meter, be it an 40-year-old film camera or a digital one has one of these! You measure the light from the position that you’re taking the picture from and the meter collects the light that is reflected from the subject into its sensor. These light meters can be highly advanced and they might measure the amount of reflected light in your whole viewfinder or just parts of it.