Clairvoyance refers to an inherent ability to foretell the future (as opposed to divining the future through signs and omens). The supposed clairvoyant may even feel that he brings about the desired course of events through sheer force of will. The clairvoyant isolates himself and meditates on the outcome, then emerges with his vision of the future. Or so he would have you believe. And some people are quite good at this. But whether or not his ability actually works (and your current, humble author does not believe that it does), presenting oneself as a clairvoyant is prohibited. The only people “licensed” to foretell the future are actual prophets, selected by God.
The reason for this mitzvah, as discussed back in parshas Kedoshim (Mitzvos #255 and #256) is that it misleads people. They think that things have power when they actually don't, causing them to turn their back on the One Who really does.
Not only may one not set himself as a clairvoyant, one may not consult such a person, or even set his mind on their predictions. (In short, no matter what it says in your horoscope or fortune cookie, you shouldn't take it too seriously.)
This mitzvah applies in all times and places. It is discussed in the Talmud in tractate Sanhedrin (pages 65a-b). It is codified in the Shulchan Aruch in Yoreh Deah 179. This mitzvah is #31 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #167 of the 194 negative mitzvos that can be observed today in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim.