Mirror and optical viewfinder
A mirrorless (ML) camera is as the name says, a camera without a mirror, nor does it have an optical viewfinder. Light flows directly from the lens to the camera sensor. The ML camera can be seen as a natural evolution of the DSLR and even the SLRs, but the ML concept is also used in cameras that do not have an interchangeable lens system, like a point-and-shoot camera or a smartphone.
The image below shows how a DSLR works. The light is sent to the optical viewfinder via several mirrors, most notably the first one that keeps the light away from the sensor (or film).
With a mirrorless camera, the light is constantly and directly sent to the sensor. The optical viewfinder is replaced by a electronic viewfinder, which basically is a small TV screen located where the optical viewfinder used to be. Not all ML cameras have a viewfinder, but instead uses the rear LCD as viewfinder.
Many DSLR cameras can flick away the mirror and show on the rear LCD what you can see in the electronic viewfinder of the ML camera. In that way, the DSLR can work as a ML when using the rear LCD.
The lack of a mirror is not to be confused with the shutter. Many ML cameras have both a mechanical and an electronic shutter. The mechanical shutter has two curtains that allow the sensor to be exposed according to the selected shutter speed by moving the curtains across the sensor. The electronic shutter turns the camera sensor on and off to get the same effect. It is all about measuring the light at the timeframe dictated by the shutter speed. Notice that for flash photography a mechanical shutter is required.
On a DSLR/SLR, the focus system is often a separate system that via a small hole in the glass uses a small area of light to determine focus or not. Therefore a DSLR focus system often only allows you to select a focus area in the center of the frame. With a ML system, the entire sensor in real time receives light and determines focus or not – this requires good software and computing power, but it also opens up for selecting focus points in the entire frame (incl. corners) and intelligence to recognize people (their eyes) and animals, and thereby take the sophistication of the auto focus systems to a new level.
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