Let's get right to the chase: two primary autofocus systems exist, contrast-based and phase detection. DSLRs use phase detection as their primary autofocus capability, while compact cameras use contrast as their sole autofocus capability.
- DSLRs — one of the uses of the mirror contained in the traditional DSLR is that it splits off some light to a phase detect sensor. Phase detect sensors are interesting, because they can report not only which direction the focus is off (near/far) but how much. Thus, in a phase detect system, the camera looks at the focus sensor, sees direction/distance and tells the lens to "move there." It's very quick, and capable of following movement very easily.
- Compacts — compact cameras use the imaging sensor for everything. They look at a portion of the image and measure contrast. At the most simplistic level, high contrast means "in focus" and low contrast means "out of focus." But a contrast-based sensor has a vague notion of direction and distance the lens needs to move, so it iterates: it moves the focus a bit and re-evaluates and then uses that information to tell the lens more about where to go.
Phase detect sensors often operate at very high data streams (usually higher in a pro DSLR than in a consumer one, which is why the pro ones are better at focus on moving subjects). Contrast-based systems operate at (usually) the video frame rate of the camera, which is generally slower.
Mirrorless cameras are an interesting breed, because we're seeing both types of systems evolve and get pushed in new directions. For example:
- The Nikon 1, Canon EOS M, and Sony NEX-5R/6 — These cameras use phase detect sensors built into the imaging sensor! That means that some pixel positions on the sensor (73 for the Nikon 1 models) are dedicated to measuring focus information, not image data. It's easy enough to hide their positions because we've got millions of imaging positions and can easily interpret the imaging data that they might have gotten.
- The m4/3 cameras — While the Panasonic and Olympus cameras are contrast-detect focus, the two companies pushed three things that were different into their mirrorless cameras compared to their compacts: (1) faster imaging frame rates for focus (now at least 120 fps); (2) faster lens focus motors; and (3) more sophisticated check-and-jump focus algorithms.
There's strong demand for mirrorless systems to have focus performance more like DSLRs than compact cameras, thus there's a lot of R&D tackling the problems right now. The m4/3 cameras were better than we were used to from contrast-based focus systems when they first appeared, and have gotten better with each subsequent generation. But they're no real match for the Nikon's phase detect system in decent light, especially for moving subjects.
Where we stand today is here: almost all mirrorless cameras have focus performance somewhere between compacts and DSLRs. The Nikon 1 cameras are close to DSLR performance in good light, a few of the non 4/3 cameras are closer to the best compact cameras in focus performance.
Overall, here's how I'd characterize focus performance today:
- Static subjects — Anywhere from good to superb. Even the contrast-based cameras are getting very good on some cameras at focusing on static subjects. There's very little focus lag when you're in Single AF mode on any mirrorless camera. Worst performer: Canon EOS M. Best performer: the Olympus OM-D E-M5 and Panasonic GH3.
- Moving subjects — Only the Nikon 1 cameras really get close to DSLR performance, and then only in good light with 1 Nikkor lenses. The m4/3 cameras have certainly improved at Continuous AF, but frankly the "miss ratio" is still too high to rely upon them for this. The laggard in autofocus is surprisingly Canon, and Continuous AF is unusable for motion on the EOS M in my opinion.
I expect these things to change (for the better) over time. But today the mirrorless cameras are pretty squarely between compacts and DSLRs in autofocus performance. Expect better than your compact camera (at least from the leading mirrorless cameras), expect worse than your DSLR (though the Nikon 1 may be an exception if you've got a very low end DSLR).