One case in which a single witness is accepted is monetary cases. Let's say Reuven says to Shimon, "where is the money that I loaned you?", and he has one witness that the loan indeed happened. Shimon, if he denies that he owes the money, would have to swear that he is exempt. This שבועה is the result of there being one witness.
Another case where one witness is enough is to testify that a ספק סוטה did indeed commit adultery - if there is even one witness, the סוטה does not drink.
Another case is in which a woman's husband goes missing, and there's one witness to his death. In this case, that is sufficient proof of death to permit the widow to remarry. In cases of עגונה, DNA evidence has been used. When sifting through the ashes of the September 11 attacks, DNA evidence was used by the Beis Din of America to permit the wives of the victims, even though no body was found.
In any and all cases of איסור והיתר, one witness is sufficient. This concept is called עד אחד נאמן באיסורים.
Regarding circumstantial evidence - it will never be used to convict someone, but it might be used to exempt someone, in the following way: Let's posit that there are two witnesses to a crime, and the witnesses have passed all the usual cross examination (דרישה וחקירה). However, the circumstantial evidence indicates that something else occurred. The judges have the right and responsibility to keep on asking the witnesses questions until the contradict each other, thereby invalidating their testimony. This in essence gives the judges the ability to utilize circumstantial evidence to throw a case out.
Last case: As @Tamir Evan pointed out in a comment above, an exception to the 2 witness requirement in crimes is in the case of murder - a king is given dispensation to execute murderers specifically with less-the-optimal evidence. The reason, in short, is that nothing is worse for society than a proliferation of murderers. It is the responsibility of the king to ensure public safety, so he has the extra-judicial right to execute murderers. For an in depth discussion of the roles of Beis Din and the King, see Drashot HaRan #11.
