Thanks to the same compromises listed above, we can also expand that "wedge" - region of apparent sharpness and detail - to a broad region by choosing a small lens opening. The effect is familiar from all of the landscape photos we see in which the depth of field extends to the closest and furthest objects shown in the scene. For now, though, let's look at ways to limit the depth of field (DOF), to confine clarity to as small an area as feasible.
The DOF will vary with magnification. The larger you make your subject within the area of the image, the shallower the DOF. This could be accomplished by simply moving closer or by using a longer focal length lens. Of course the longer focal length will mean a narrower angle of view is included in the picture. Although an object in the background might be equally blurred in a wide angle shot and a telephoto shot, that object will cover a larger area in the telephoto version, so it will look "softer".
Sometimes overlooked when considering DOF, but often of equal or greater significance, is the character of the blurred areas outside the region of sharp focus, and also the transition from crispness to softness. This will vary with different optical designs.
This first example photo was shot in response to a friend's comment about galactic images that were modified in Photoshop to create artificial transitions from in-focus to out. I did not do any software manipulation of this image. I used a tilt-shift lens (24mm), and instead of tilting the plane of focus to accommodate the "galactic" spray of leaves, I tilted it in the opposite direction, so that it intersected just a small spray of berries in the mid-ground. This gave a nice soft character to the foreground leaves and a reasonably soft background for a wide angle lens of middling aperture (f/3.5).
For this carved African bird, I used a more typical approach to the hunt for shallow DOF - a somewhat longer focal length (85mm) and a very wide aperture (f/1.2). Your attention is called to the eye, and what could have been a very distracting background becomes just some soft colors.