Historically, digital cameras have used an angled mirror to bounce light coming into the lens into a pentaprism and through an optical viewfinder. This meant that the photographer was looking at the light from the scene directly. It also means the camera physically requires room for the mirror and that the mirror mechanism is one more part subject to failure.
Mirrorless cameras do away with the mirror. This means you use an Electronic Viewfinder or EVF and are looking at a little LCD screen’s representation of the image rather than at the actual scene. This means you see exactly what you’re shooting (since you’re looking at a digitized version of the analog world) but it also means your battery has to power that LCD continuously.
Early mirrorless cameras had poor battery life, subpar autofocus systems, and poor resolution viewfinders which blacked out when a photo was taken. Despite many photographer’s reluctance to convert, mirrorless is quickly becoming the future of photography. As battery life on cameras improves, the advantages of mirrorless cameras are simply overwhelming the traditional DSLR (mirrored) cameras.