Aspherical Elements

Aspherical elements are meant to reduce the level of distortions and to improve the sharpness at the image borders. Basically aspherical elements can optimize the focus of edge and center light rays (see picture below). It is very popular to name a lens e.g. "28-80 aspherical" in these days but the word "aspherical" does not reveal the effectiveness and quality of the aspherical element.

There're three types of aspherical elements:

1. grounded
2. molded
3. hybrid

Hybrid asphericals are quite cheap to produce and widely used for cheap consumer grade zoom lenses. Basically it's a spherical glass lens with an additional plastic surface which forms the aspherical shape.
Ground aspherical lenses consist of glass only (obviously). The grinding process of a spherical element and the following polishing process is very expensive so you only find them in Pro-class lenses (e.g. Canon 17-35L, Nikkor 20-35 etc.) and a few older lenses.
Molded aspericals are produced by directly molding glass in a molding machine by incorporating an aspherical metal die. These elements are often used in good amateur class lenses.

The gain of quality by using hybrid lenses seems to be quite limited while lenses with true ground aspherical lenses often produce outstanding results, molded aspericals are in between quality-wise.
Apart from the sheer material and production aspects the positioning is also very important. The best wide-angle lenses use a large diameter aspherical element in the front group (probably for a better correction of light rays with steep entry angle ?), rear aspherical elements (=smaller and therefore cheaper) are less effective.
However it should also be mentioned that the maybe very best wide-angle zoom, the Carl-Zeiss T* 28-85/3.3-4, doesn't feature any aspherical elements at all so the game plays also in the quality of the general lens construction (that should be obvious anyway).
In general asperical elements are used in ultra-wide and wide-angle fixfocals & zooms as well as standard and so-called super zooms.

"Apochromatic" Elements

"APO" elements (UD, SUD, CaF2, LD, SLD, ED etc.) improve contrast and sharpness by reducing chromatic aberration (color defects) that usually occur in tele lenses. These elements are able to focus different wave lengths of one light ray in one point (see picture below). These elements are quite expensive and usually not used for cheaper lenses. The problem is however that the quality of these special elements varies heavily so the effect is often downgraded to a marketing gag - this is especially true for some third-party manufacturers!
As a rule-of-thumbs a good long tele lens will always feature two or more of these special elements. Recently the first ultra-wide and wide-angle lenses emerged using APO elements besides aspericals in order to reduce problems with lateral color shifts.