There is no important difference in image quality between mirrorless cameras and DSLRs.
Both types of cameras print equally well at 4x6 and 8x10.
Both types of cameras produce images that display equally well on the web.
You can even throw high-end compact cameras into the mix, and say that high-end compacts are just as good as mirrorless cameras and DSLRs in most situations.
What differences there are in image quality, mostly control over depth of field and noise at high ISO, will matter a lot to some people, some of the time, but not to most people, most of the time. Click the tabs above for depth of field and ISO if you are not sure why control over depth of field and noise at high ISO are important.
The picture of the pelican below was taken with my Olympus C7070 7 megapixel 2005 compact. The image quality is not as good as I would have gotten using the best mirrorless cameras and DSLRs available today, but it is awful close. And the C7070 is a dinosaur compared to today's technology. Most of us should not be losing sleep over image quality. We are living in the Golden Age of photography. All of these high end cameras do a great job.
The Cameras You Want for Ultimate Image Quality
If you need the absolute highest image quality possible, these are my recommendations.
The Nikon D3X has the best image quality of any camera today, period. This is a pro's pro DSLR, with extremely high image quality, and professional handling and performance. If you are producing prints for museum galleries, this is the camera to get. Not far off is the Canon EOS-5D Mark II , Canon's top of the line professional DSLR.
The Sony Nex 7 has the best sensor, and therefore the highest potential image quality, of any mirrorless camera. The image quality of top micro four thirds cameras, such as the Panasonic G3 and the Olympus E-P3, is close to the image quality of the Nex 7. Perhaps more importantly, the micro four thirds cameras benefit from a more developed lens line than Sony cameras, allowing you to adjust your lens to a wider range of shooting conditions.
The Olympus XZ-1 has the best image quality of any compact camera. Olympus had a history of making excellent high-end compact cameras before abandoning their well-received C line of cameras. With the XZ-1, they have gotten back in the mix. Other top compacts are the Panasonic LX5, and the Samsung TL500.