A Mirrorless Digital Camera is a Digital Camera that accepts different lenses but does not have a mirror and mirrorbox to switch the scenes inside the camera from the optical viewfinder to the imaging sensor and shutter. Instead, most viewfinder functions are electronic. This allows the camera body to be smaller, thinner and lighter. These are also known as MILC cameras - Mirrorless Interchangeable-Lens Camera.
In many cases, these smaller bodied cameras also fall into the category of 4:3rd cameras (Four/Thirds Cameras) or Micro 4/3 cameras. This refers to the shape and dimensions of the actual image sensor (4:3 instead of the traditional 3:2 layout) and allows for many different camera systems to use each other's lenses. Below are the Mirrorless and 4:3rds options from FujiFilm.