What is a light meter?
How do I use one?
What sort of photography will fool a digital camera's meter? Hence for those you do need to meter.
A light meter is a device that measures light.
There are two types of light meters:
All digital cameras have a reflected light meter built-in. Actually, they have several, up 1200 of them! If you put your camera to Spot metering mode, then it measures light from one area (between 2%-10% of the frame, configurable on some higher-end Nikon DSLRs) which is what hand-held reflected light meters do too.
This is how a digital camera measures exposure. Hence, there is little use for a stand-alone reflected light meter for digital photography. However, the read-value is highly dependent on the subject, since dark/less-reflective subjects reflect less light than light/more-reflective ones.
An incident must be placed very close to your subject, so that it receives the same light. Once you do that, the reading is good for that ambient light, no matter what color or how reflective is your subject. These meters are typically used in studio environments where light is setup for portraits or product photography. You only need to meter when the lighting changes.
Since these devices are separate from your camera, you have extra work to do to use them. First, you usually need to input the ISO and sometimes aperture or shutter-speed into the meter. When the reading is taken, the meter gives you the missing parameter, usually shutter-speed or aperture. You put these values into your camera in Manual exposure mode.
A light meter is a (usually hand-held) device containing a light receptive cell and some sort of display. When pointed at a scene it will take a light reading and show you the appropriate camera settings required (aperture, shutter speed and film speed) to get a correct exposure. These days, light meter apps are available for most smartphones (using the built-in camera as the light cell).
You can usually lock down one or more of the settings: for example, if you're using a film camera with a 400ISO film loaded, you'd lock the ISO setting at 400 so it doesn't unhelpfully show you settings for a different film speed.
Light meters are useful when using a film camera without either automatic metering or a built-in light meter. I can't think of any use for them with a digital camera: all digital cameras have automatic metering. Even under unusual lighting conditions it's much easier to take a shot, check the screen and try again than fiddling about with a separate device.