I think Dive-aholic gave the most commonly accepted definition: Technical diving is diving under a virtual or real overhead. That means cave diving, wreck penetration, or incurring sufficient decompression obligation that a direct ascent to the surface is no longer an option.
Because of the risks incurred in putting yourself in a position where you CAN'T surface if you have a problem, most technical diving is done with some kind of redundant system, which most commonly means double tanks connected by some kind of manifold. Double tanks are generally used by bolting them to a rigid plate with a harness attached to it, rather than a soft, fabric BC. In addition, people who dive in overhead environments usually have a long hose attached to one of their regulators, so that they can exit through a narrow space while sharing gas. I assume this is the setup the guys in your LDS showed you.
Because of the risks incurred in putting yourself in a position where you CAN'T surface if you have a problem, most technical diving is done with some kind of redundant system, which most commonly means double tanks connected by some kind of manifold. Double tanks are generally used by bolting them to a rigid plate with a harness attached to it, rather than a soft, fabric BC. In addition, people who dive in overhead environments usually have a long hose attached to one of their regulators, so that they can exit through a narrow space while sharing gas. I assume this is the setup the guys in your LDS showed you.