Crystals grow in three dimensions. Terminations are the "ends" of a crystal which has equal or similar growth patterns in two dimensions but not in the third. The growth in that one dimension then is typically much shorter or longer than the other two. Any crystals in the cubic system do not have terminations, because the crystals grow equally in all dimensions. Garnet and Pyrite are two examples of this system. So terminations show up in crystals of the trigonal (like tourmaline, low quartz), hexagonal (such as beryl), tetragonal, etc. A long skinny crystal could be called "acicular" and a short squatty one would be tabular. The ends of either of these would be considered terminations.
acicular tourmaline and squatty beryl.
If a crystal grows unobstructed into some space, the end is considered a "growth" termination. These naturally facet as do the sides, but with different rules. With Elbaite tourmalines, the most common "growth" termination to look for is the plain pinacoid. It's a flat-topped crystal, with signs of active accretionary growth. In the above Sapo blue-cap, you can see the evidence in the triangular growth patterns apparent atop the mirror-like flat pinacoidal surface. Similar growth patterns can be seen on the highly reflective surface of the aquamarine shown here as well, though not triangular.
Commonly though, a crystal will show "contact" terminations, where the crystal growth was obstructed so the natural growth terminations could not appear. The crystal grew against some other object and cannot naturally terminate. In essence, what you see is the imprint of the surface upon which the crystal grew. In a contact terminated crystal, you can often see large crystal imprints, or more typically just a rough surface. But there are no conchoidal fractures or other signs of breakage. Classically, Hot Springs quartz will have rhombic gaps in the base where the basal calcite has dissolved away. This is a most extreme case of contact termination, with often beautiful results. With tourmalines, you will frequently see one end growth terminated and the other termination contacted or broken.
Here's a tourmaline with a pinacoidal termination that was half contacted. The left side shows highly reflective but slightly bulbous surface. The right is what remains after a contacting cleavelandite layer was removed. Even though natural, the value of this piece is diminished by the flawed termination. A broken end is not a true termination, unless it has healed naturally. In tourmalines, healed breaks occur quite frequently. The repairs may glue the crystal back together, or result in a doubly-terminated crystal.
A doubly-terminated tourmaline. The left side is the original growth termination and the right side is a healed break that became a growth face.
The base of this crystal is broken on one side. (Photograph by Brian Kosnar)
With Tourmalines, things can get a little more complex. The end you see above is in fact a growth termination. Commonly called a "Crow's Feet" termination by miners familiar with the material, it consists of a collection of small individual trigonal terminations, and all part of the same crystal. The overall appearance is rough, but when you rotate the end around in good light, you see the individual reflections showing the composite termination. This same kind of behavior can be viewed in corroded crystals as well. Not always extremely beautiful, but at least natural, and some of the most famous and valuable tourmalines have this kind of termination.
What you don't want to see is this: A crystal advertised as terminated, but on inspection you can see saw marks that indicate this crystal has been cut. This is not a natural termination, and the specimen value is greatly reduced. This is not a piece for the collector of natural crystals. Note the irregularities at the start and end of the entire cut - also indicative of cutting. If the end was polished, you may not see the cuts so clearly. But as in the case of tourmaline, you will not see any natural growth patterns. Any termination that does not have the natural irregularities of a growing crystal is suspect.
Where it gets hard is when the specimens are sand-blasted after cutting. You end up with a rough but believeable termination. But in fact, if you have a rough termination, and you can't see the facet reflections in good light, there's a good chance the stone has been modified.
Just as an addendum, side facets in elbaite tourmaline are typically striated because of the complex growth pattern they exhibit. A healed break looks like a cross-section with whitish, almost pearly "cats-eye", probably because of the intergrowth from both sides that don't quite line up. You should be able to distinguish this relatively easily from a reconstructed patch. I'll go into more detail on this in another guide.