The five string banjo is a folk instrument essentially, and so, like most folk instruments, it can be tuned and played many ways. A great many different tunings for the banjo have been devised. Even though this is the case, in modern times, banjo-pickers usually only play in the open G tuning, occasionally changing to D if they play bluegrass (though even that is becoming more and more uncommon) or C if they play in a more folk style, such as clawhammer.
I've tried to indicate which note each string makes in as simple a way as possible. The first letter corresponds to the fifth (or drone) string. It is represented by a lowercase letter. The drone is always in the octave two octaves above middle C. The other four are represented by capital letters. A letter with the number 1 after is in the octave above "middle C", a letter with 0 after is in the octave below.
G major, "open G", or "high bass" is by far the most common tuning. It's the tuning used in Bluegrass, and some folk songs. g D0 G0 B0 G1
C,"low bass," or "classic C" isn't quite as common, but is still widely used in folk music. Like Open G, it is a survival of an old tuning from the nineteenth century/minstral era, but up a whole fifth from the original. There are other tunings to play in the key of C, which at times are more practical, however this tuning has the advantage of being easy to switch into and out of if your banjo is normally tuned to G. It also shares the distinction of being one of the earlist konwn banjo tunings with open G. g C0 G0 B0 G1
"Double C" is a very popular tuning with clawhammer players. It is nearly the same as classic C, except that the B string has become a C. g C0 G0 C1 G1
"Mountain minor," "G-modal" or "Sawmill" is a variation on open G used in old folk songs of the Appalachians. The name is somewhat misleading, considering that it's actually a Gmaj6 chord, rather than being any sort of minor chord. g D0 G0 C(middle c) G1
D major (open D) is another folk tuning, occasionally heard in bluegrass. a (f# is also common) D0 F#0 A0 D1
G minor is yet another folk tuning. g D0 G0 Bb0 D1.
Open C was used by the legendary Uncle Dave Macon on many of his songs. It has several technical advantages over the normal C, however it is harder to tune into from open G. g (or a) C0 G1 C1 E1
How to Play the Five String Banjo by Pete Seeger
My own banjo-picking