Turn to the big A (or Av) on your camera's mode dial when you want to control background blur and don't care about much else. Here's some guidance on how to use Aperture-priority.
As you might expect, you use Aperture-priority mode when you need to control the size of the opening through which light travels to reach the sensor -- usually to affect the depth of field -- but don't care (much) about shutter speed. You set the aperture you want and the camera automatically adjusts the shutter speed to maintain the correct exposure.
Which raises the question: What is depth of field? It's the perceived area of sharpness between near and far objects. Shallow DOF means there's a very small area of sharpness (i.e., the subject is in focus and the background is relatively blurred). Some cameras now offer a Background Defocus effect to simulate this. Deep DOF implies that near and far objects are both in focus.
For more information about other camera modes, check out " What are all those camera modes for, anyway?"
Anybody with a mode dial on their camera, physical or virtual, dSLR, interchangeable-lens mirrorless, advanced compact or even a phone with manual controls.
When should you care most about aperture?
I think the last situation is really the only compelling use of Aperture-priority mode, but your mileage may vary. Note that you should obviously care about aperture when it comes to shooting in low light or bright light, because widening the aperture lets in more light and narrowing it lets in less. However, I find Aperture-priority mode is a less optimal solution than Shutter-priority mode in those circumstances, for reasons I cover in the Drawbacks section.
You set the aperture and the camera calculates the rest. However, that statement begs the underlying complexity of the relationship between aperture and DOF. Unlike shutter speed, which has a pretty understandable cause and effect in your photos, the effect of the aperture setting as it affects DOF depends on your camera's sensor size, the focal length setting of the lens, and the distance you are from the subject.
Take, for example, the two photos above. Shooting close to Shirley's face blurs out the background nicely, but backing up to get both Shirley and Ramona in the frame results in a far less blurred background -- despite the fact that the first photo has a wider aperture than the second. These were shot with the Canon EOS 7D , which has an APS-C-sized sensor; on a camera with a larger full-frame sensor the blurring would be more pronounced in both photos, while with the smaller sensor of a Micro Four Thirds ILC the background would be sharper in both shots. (You can try different values in a DOF calculator to see the varying relationships, and there are similar calculator apps for mobile devices.)
As an aside, this is why you should always take claims about wider apertures proving shallower DOF in phone cameras and low-end point-and-shoots with a boulder of salt. The sensors are so small that the affect of the aperture is trivial compared with distance from the subject.
The other factor for shooting in Aperture-priority is the hard limit on how high or low a value you can choose, determined by the lens. Those kit lenses that ship with entry-level dSLRs and ILCs tend to have a maximum aperture of f3.5, and that's only at their widest angle of 16mm or 18mm. If you're constantly frustrated by that aperture limit, you may need a better lens.
Some cameras use A to designate Aperture priority on the mode dial, while old-fashioned models call it Av (for Aperture Value). You may need to press a lock button to turn the mode dial; if there's no physical dial, usually more common with ILCs and advanced compacts than dSLRs, then you usually pull up the mode settings via a quick-menu or function button. If you're not sure, then you'll have to consult the manual.
How to read the settings: At least this part is easy. On a camera that has a quick view screen it's usually the number next to "F" on the display. ("F" stands for "F-stop" or "F-number", calculated by the lens' focal length divided by the diameter of the aperture.) If there's no "F" it's usually the number next to the shutter speed on the display, and will likely show values between 1.8 and 32. If they're outside that zone, you probably have a really nice lens and already understand this stuff. Smaller numbers mean wider apertures, larger numbers mean narrower ones.
Changing the value: On cameras with two dials, usually one on the front and one on the back, different manufacturers use different conventions for the primary adjustment dial. For instance, Canon uses the front dial to adjust aperture value in this mode while Nikon uses the back dial. Lower-end cameras generally have just one dial and point-and-shoots sometimes use the navigation buttons.
Choosing an aperture: Keep in mind that if you have a zoom lens with a variable aperture range (denoted as say, an 18-55mm f3.5-5.6 lens rather than 24-105mm f4), the widest aperture will decrease as you zoom in to the telephoto end -- e.g., go from 18mm f3.5 to 55mm f5.6 -- and reverse as you zoom out to wide angle. For everything but studio-type work, if you're going to shoot in Aperture-priority mode with an inexpensive kit lens and want the closest you can get to a set-it-and-forget it choice, I vote for f5.6, at least in good light. That will ensure snapshot-quality sharpness of most things you plan to shoot, and will keep the aperture from changing as you zoom. An alternative is to set it f3.5 (or whatever the widest your lens supports) with the understanding that it will change automatically as you zoom, but it will automatically change to be set to the widest aperture possible for a given focal length.
If you want maximum sharpness throughout the scene and there's plenty of light, then f8 or f11 is a good choice. Try to stay away from f16 or higher on inexpensive lenses and small sensors, since sharpness tends to decrease past a certain point as other laws of physics intrude.
If you have a fast lens that supports apertures of about f2.8 or wider, there are some caveats to shooting wide open. First, the wider you go the harder it is to focus accurately; the smaller the zone of sharpness, the more difficult it is to keep the camera fixed on the appropriate point. This is especially true if you're depending upon autofocus. Also, cheap, fast lenses, like a typical 50mm f1.8, tend to produce fringing on the photos at their widest.
You can usually figure out within the first few shots what setting produces the image you want. Just remember:
Increasing the aperture number setting narrows the aperture and broadens the area of sharpness for a given focal length and distance from the subject.
Decreasing the aperture number widens the aperture and shrinks the area of sharpness for a given focal length and distance from the subject.
"A" doesn't stand for "Auto," but you can stick with many of the automatic defaults in Aperture-priority mode. You might want to change these if you're still unhappy with the results, though.
I have to admit: I'm not a big fan of this mode and don't use it much. If I want to control aperture I tend to jump straight to manual mode; I figure that as long as I have to think a lot I might as well go all the way.
In a controlled, well-illuminated environment with little to no movement -- in other words, cases where shutter speed really doesn't matter -- Aperture-priority is a fine choice for controlling depth of field. However, under frequently changing lighting or in dim conditions, you run the risk of the shutter speed dropping lower than that which you're comfortable shooting handheld. Unlike Shutter-priority mode in which the camera will simply underexpose if it can't open the aperture sufficiently, in Aperture-priority it will keep dropping the shutter speed (unless you've set a limit). If you don't pay attention, or frequently review your photos, you may not realize you've been shooting at 1/3 second. Underexposure is somewhat fixable; completely motion- and shake-blurred photos are not.