Taking a clear portrait photo with blurred background draws the focus to what’s important. This is a popular technique for isolating subjects from backgrounds. The term that refers to the quality of the blur is "bokeh."
There are two main factors: selective focus and shallow depth of field.
1. Selective Focus
To selectively focus on one particular subject in the scene, limit your autofocus system to use a single autofocus point, then select the one autofocus point you want to use. In some cameras, you may already be limited to one point in the center.
Align the active autofocus point over the target and lock autofocus. In most cameras, the default to do this is to press the shutter release button halfway down. Many photographers prefer to assign this function to a separate button on the back of the camera instead.
With the autofocus locked on the subject, and without re-engaging autofocus, re-align your camera until the shot you want is framed in the viewfinder. Then press the shutter release button all the way down to commit the shot. This technique is called "focus and recompose".
Alternatively, you could manually focus where you want.
This should bring your focusing target into the best focus. And because of the way focus works, anything else in the scene at the same distance away from the camera is going to be at equal focus.
2. Shallow Depth of Field
Depth of field is the range of distances nearer and further than your focusing distance where things also appear within acceptable focus. If you selectively want a subject at a certain distance in focus but not other things at other distances, you want to reduce the size of this range, or a shallower depth of field.
Several factors will affect this:
Aperture: In addition to being an exposure control, the size of your aperture is also frequently used to control depth of field. A larger aperture (smaller f-number) will make depth of field more shallow while a narrower aperture (larger f-number) will make depth of field larger. Your aperture size has a maximum dictated by the design of your lens, and cheaper lenses often do not have very wide maximum apertures available; a wider-aperture lens may be necessary for the effect you want.
Focusing distance: As you focus closer, the depth of field decreases. The effect may be easier to achieve on very close subjects.
Focal length: As you zoom in or otherwise use a longer focal length, the depth of field decreases. Perhaps more importantly, the compression effect of the longer focal length will enlarge the background to a greater degree than it enlarges a closer subject. An enlarged background also enlarges the blur seen in an out-of-focus background, which will make the effect appear more pronounced.
Format size: A larger format sensor or film frame captures a wider field of view at a given focal length compared to a smaller format. Thus, assuming you are comparing between the same field of view, the larger format is going to require a shorter distance and/or longer focal length and will therefore have a shallower depth of field. This is why shallow depth of field can be difficult to achieve with small format phone and point & shoot cameras, and easier to achieve with full frame or medium format cameras.
Background distance: This does not affect depth of field. But the further away your background is relative to the subject, the further outside of the depth of field it is, and the more blurred it will appear.
A blurred background is created through a mix of gear, aperture, and subject placement. You can combine all three to create the most background blur.