The courts are forbidden to pass judgment based on the testimony of a single witness, no matter how great a person that one witness might be.
The reason for this mitzvah is that a single witness might be unreliable. Not only might he be in error, even a great person can give into temptation and fall. It is not inconceivable that even an honest person might bear a secret grudge and testify falsely against someone who had wronged him. We absolutely need the safeguard of a second witness to eliminate this possibility, even if it is remote. (Even if the defendant is an absolute scoundrel, he still deserves a fair trial!)
This law only applies to imposing penalties in criminal matters. In a civil trial, where it's one litigant's word against the other's, an oath is imposed based on one witness testifying on behalf of one of the parties. In matters of forbidden items, a single credible witness is believed, such as to tell you that something is kosher to eat.
The criminal aspects of this mitzvah only apply at a time when the courts can impose physical penalties; the ramifications for civil cases apply in all times and places. This mitzvah is discussed in the Talmud in tractate Shevuos (40a), in Sotah (2a-b) and elsewhere. It is codified in the Shulchan Aruch in Choshen Mishpat 28. This mitzvah is #288 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #73 of the 194 negative mitzvos that can be observed today in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.