680 means 68 x 10^0 = 68 x 1 = 68.
Also, it's not strictly incorrect to refer to a capacitor as a tantalum electrolytic, but since all tantalum capacitors are electrolytic, calling them "tantalum capacitors" is sufficient and pretty much universal.
In general, where you know what the dielectric is, that is what you call the capacitor. So we have ceramic, paper, polyester, mica, vacuum and air capacitors (among many others)
But wait you say. How can tantalum be the dielectric when it is a conductive metal? Well, in the case of electrolytic capacitors, the name is given as the material on which the dielectric is formed (often an oxide).
Is it this simple? Of course not! The common electrolytic capacitor uses an aluminum anode with the dielectric formed from aluminium oxide. These are generally just called electrolytic, or aluminium electrolytic, and rarely just aluminium. A newer type of electrolytic is called a polymer capacitor. These are electrolytic capacitors with some unspecified anode material and a conductive polymer used as the electrolyte. Also, ceramic capacitors may be called multilayer ceramic capacitors (MLCC), or even a funny code like X7R which refers to the characteristics of the ceramic used.
Sometimes the name given is based on the construction (e.g. feed through capacitors), or function (bypass capacitors). Both of these are typically ceramic, but you'd never know
So basically, there's a mix of history, convention, and exceptions here. Don't sweat on getting it right all the time.
Things get were with other components. In the early 60s, if you just said "transistor", it could be pretty much assumed you meant a PNP germanium transistor. These days, you would assume an NPN silicon transistor (although other materials like gallium arsenide and silicon carbide can be used). And even then, PNP or NPN implies a bipolar transistor, not
junction field effect transistor (jfet), or an insulated gate field effect transistor, these days usually named after the meeting of insulation (mosfet) -- oh and we assume an enhancement rather than depletion mode device. Oh, and is the bjt epitaxyl, point contact, or one of the many other mostly historical methods of construction?
Aaaaaaaaaaaagh