Early designs were TLR or rangefinder cameras, both of which did not look through the lens used to expose the film. This created a number of problems which were fixed with the advent of SLR, which employs a mirror and prism designed to allow the photographer to look through the lens used to create the image. The disadvantage to SLR was it required an increased distance between the back of the lens and the media to allow for the mirror. This means all things being equal, the lens has to be larger and heavier for the same image ratio.
Mirrorless cameras do not employ a mirror and instead have a viewfinder that depicts a representation of what the digital sensor sees. There are several advantages the most noticeable being it allows for smaller and lighter lens designs, like the old rangefinder cameras. It is also less mechanical and more quiet(important for some nature photography) because you don't have a mechanical reflex mirror operating inside the camera. Some mirrorless cameras still have a mechanical shutter, so the noise hasn't been completely eliminated, but others have an electronic shutter which allows for completely quiet operation.
Early mirrorless cameras had problems with shutter delay which has pretty much been completely resolved with faster processors. Another disadvantage is you no longer have an optical viewfinder which some photographers are going to want, particularly if they are used to optical viewfinders.
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