Many lenses (like the Pentax DA 15mm f/4 Limited, or the Nikon AF-S 35mm f/1.4G) are described as having "aspherical lens elements". Does this mean that regular lenses are spherical lenses? What's the difference, and what are the advantages of one over the other?
The photographer's answer is that it does not matter. We let optical engineers find the best way to build lenses and then use them for our craft.
The technical answer is yes, non-aspherical lenses are spherical in the sense that one of their surfaces corresponds to the outer surface of a sphere of some radius.
Aspherical lenses are more complex and are not constrained to following the curve of a sphere. That also means there are lots more variance among aspherical lenses. This gives optical engineers more freedom when designing such lenses and therefore more chances to correct for optical issues.
It is simple and relatively cheap to make lenses whose surfaces are parts of a plane or parts of a sphere. Such lenses do not focus light perfectly; this lack of focus is spherical aberration. This graphic from a Wikipedia article schematically illustrates how the light rays fail to converge (lower half) compared to a hypothetical perfect lens (upper half).
Spherical aberration is especially prominent in large, fast (bright) lenses. This lack of focus shows up as various forms of fuzziness. The problem can be corrected by placing other lenses in front of and behind the lens. It can also be corrected by changing the shape of the lens surfaces (making them aspherical), but that tends to be more difficult and expensive to carry out: glass spherical shapes are simple to make and measure; aspherical ones are not.
When additional lens elements are added as correction, they increase the amount of flare and reduce the contrast of the photograph, as well as adding to the size and weight of the lens. They might also alter the colors a little. Therefore, aspherical elements typically signal a lens that is crafted to produce contrasty, colorful images and to be lighter and easier to use. They do not in themselves assure high quality, because lenses can exhibit other problems besides spherical aberration. For instance, small aspherical lenses can be molded of plastic and routinely appear in cheap photo equipment. However, a large glass aspherical lens is much more expensive to produce and usually reserved for better lenses.
A lens marketed as "aspherical" will usually have only one surface (one side of one piece of glass) aspheric, and all the other surfaces will be spherical (or flat).
So the vast majority of glass in any lens, even those lenses marketed as aspherical, is spherical anyway.
An aspheric surface can help correct for spherical aberration, which can make the image look soft when the aperture is wide open. But it's not an exact science, because compromises have to be made. For example, the spherical aberration correction can sometimes make background bokeh more harsh, which isn't great for portraits.
Aspheric lens surfaces are much more expensive to produce, because they can't be ground by a natural rotary motion.
Fun fact: high quality aspheric lens surfaces have been around since 1667, used in telescopes, reading glasses and burning glasses (!).