You are desiring a photo that has "shallow depth of field"; this is also known as "selective focus"; this is sometimes referred to as a blown-out background, but not blown-out in the sense of over-exposure, but blown out meaning thrown greatly out of focus. Achieving this type of shallow depth of field, shallow enough to create a really "blown-out background" is easily achievable with a wide lens aperture, combined with a camera that is physically close to the subject; and a subject which is close to the camera but while is FAR IN FRONT OF the background.
Physically being CLOSE TO the subject is the most important, or easiest way to get this effect. With a close subject and a backdrop that is very far away from the subject, it's pretty easy to get this kind of effect.
A long focal length lens that ALSO has a pretty wide aperture, like say from f/1.8 to f/2.8, can also help in achieving this look.
With a compact P&S digital, a subject at 3 to 5 feet, and a distant backdrop, this kind of effect can be achieved with longer lens settings. The shot of the woman looks to me like a 300mm lens, used from fairly close, and an aperture of maybe f/5.6 or so. I can see some people, recognizable "as people"; with a 300 f/2.8 lens on FF, the people would just be diffused, color blobs.
Here is a Nikon FX frame, full image, made from 5.01 meters, at f/5.6, at 1/1250 second at ISO 400, using the 260mm zoom setting on the 70-300 VR-G lens. This is a distant, 1/16 mile or so distant, forested background, which is "blown out", focus-wise.
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traveloregon_D3X_7792_Aelisa.jpg photo - Derrel photos at pbase.com ]
And here is a shot from the SAME exact distance, but the lens zoomed shorter, to 75mm, with the aperture also at f/5.6 at 1/1250 second. See the difference?
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traveloregon_D3X_7894.jpg photo - Derrel photos at pbase.com ]