SLR-quality in a compact package
'Mirrorless' camera is a bit of an unfortunate term in that it defines the tool by what it lacks: the moving-mirror found in every DSLR. Mirrorless cameras (more accurately called mirrorless interchangeable lens cameras) do retain one attractive feature of DSLRs: their interchangeable lens mount. By discarding the mirror, but keeping the interchangeable lens, mirrorless cameras can provide a cheaper, smaller package with image quality and shooting flexibility that can rival their older DSLR uncles.
If the DSLR is the older, larger uncle to the mirrorless camera, the compact camera is the diminuitive cousin: these cameras are typically much smaller than the DSLR and mirrorless camera. This size difference is due to compact cameras' lack of an interchangeable lens and their generally smaller sensors. With the compact camera, you can get somewhat smaller, cheaper cameras at the expense of shooting flexibility, low-light performance and general image quality.
It's important to note that the above is a generalization, especially since the lines between camera categories are blurred more today than ever before. The Sony RX1 provides an SLR-sized sensor with an exceptional lens in a compact camera package, while the Canon SL1 provides a DSLR experience in a (near) mirrorless-size package.
It's also worth noting that you may hear other terms used to describe mirrorless cameras, including 'Compact System Cameras (CSC)' and 'Electronic Viewfinder Interchangeable Lens (EVIL).' Recognizing this flexible terminology, we also have a compact system camera page.