What's a mirrorless camera? Well, I mean, it's obvious what it is, but why is it good to not have mirrors inside cameras?
Primary benefits: mirrors exist in SLRs to allow the photographer to see what the lens sees: wide angle, normal focal length, telephoto, etc. Combined with a pentaprism, the optical viewfinder displays the image right side up and not backwards, the same as how your brain interprets what happens in your eyes.
It also allowed for things like helping the photographer manually focus (typically using a split prism microcollar focusing screen) and depth-of-field preview.
Primary cons: first of all, mirrors are slow. There's a delay after pressing the shutter button because the mirror has to get out of way; it flips up before the focal plane shutter curtain opens.
They're also mechanical and can be easily damaged since they are quite fragile (lightweight mirrors respond faster than thicker ones) and dangerously close to where a photographer's fingers are when he/she is changing a lens.
They also cause vibration. High-end film SLRs had mirror lockup levels to reduce the vibration caused by the mirror flipping up.
The mirror also causes major compromises in lens design. Ideally the rear lens element is as close to the focal plane as possible. The mirror in an SLR gets in the way so the rear lens element has to be in front of the mirror. This forces optical lens designs to put the focal plane far behind the lens. This is referred to as retrofocus. In order to do that, the lenses need to be bigger which means heavier and more expensive and often have more elements which additionally means more distortion, more lens flare (non-image forming light rays) and less light reaching the recording medium (each uncoated glass-air surface reduces light transmission about 5-10%).
To reduce image distortion sometimes lens manufacturers would introduce aspherical elements in their lens designs. Aspherical lens elements are more difficult to manufacture and thus add to the cost.
Without the reflex mirror, the lens designer can create more optimal lens designs with rear lens elements very, very close to the focal plane. This also means relatively smaller lenses often with fewer elements. If you look at a 50mm f1.4 lens for a 35mm rangefinder camera like a Leica versus a 50mm f1.4 lens for a Canon EOS 35mm SLR, the Canon lens is huge.
Many of these ideal mirrorless optical lens designs were perfected about 120 years ago (late 19th century/early 20th century) in the era of view cameras -- before the SLR's invention.
Modern digital cameras have LED viewfinder screens with lots of detail, largely eliminating the necessity for an optical viewfinder and hence the reflex mirror.
