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Disclaimer: There are no definite rules here; everything I'm about to say is a generalization to which many exceptions occur.
The most common types of banjos being played today are the 5-string, tenor, and plectrum banjos; there are not only physical differences among these instruments, and different ways of playing them, but the types of music for which they're used tend to be different, despite some notable overlaps.
The most commonly played banjo in the United States is the 5-string banjo. Obviously, these banjos have five strings; what is not obvious from the name is that the 5th string is a shortened string that most commonly is attached on the side of the neck, as shown in the whole-banjo photo, below, left. The close-up (below, right) shows a geared 5th string tuning machine protruding from the side of a neck. This string is usually tuned to the highest pitch, yet is adjacent to that tuned to the lowest pitch. The fifth string is not often fretted, and so it usually functions as a "drone" string: It is this drone that gives the 5-string its unique character.
There are many tunings used for the 5-string banjo; on this website is a compilation documenting over 120 known tunings! Despite that huge number, the most commonly used tuning is an "Open G" tuning, in which the open--or unfretted--strings sound a G major chord: gDGBD (the indicated string order is 5,4,3,2,1-- left to right in the accompanying picture--and the lower case "g" indicates the short, 5th string's being tuned an octave higher than the 3rd string--the other "G"). You can hear this tuning, string-by-string, on Harrell Stiles's Banjo Page.
There are many styles of 5-string banjo, and an almost equal number of uses for them. Bluegrass music commonly uses the 5-string (for many, it is the 5-string that defines bluegrass). The style of playing entails picking upward on the strings with the pick-covered middle- and index-finger tips and downward with the thumb, also bearing a pick. This "3-finger style" can be heard on The Banjos That Destroyed the World in this recording of Earl's Breakdown. A review of this and other albums (with many more soundbytes) can be found on the Cybergrass website, as can a lot of general bluegrass information.
Many folks play Celtic music on the 5-string banjo. While he plays a mean bluegrass banjo, Chris Grotewohl also plays Celtic tunes in 3-finger style, which you can hear on his recording of Paddy Fahey's Jig. Visit his website for more sound files and to purchase his recordings and book.
Cathy Fink plays a very smooth clawhammer style on her recording of Banjo Song. You can hear more of her work and buy recordings at tunes.com.
Dwight Diller has a unique approach to clawhammer; listen to him play Cluck Old Hen. There is a Dwight Diller website from which you can download tablature, hear more of Dwight's work, and buy recordings.
The Tenor Banjo and the Plectrum Banjo are four-stringed instruments. The Plectrum Banjo has the same length neck as a typical 5-string, but--of course--doesn't have that weird, short, fifth string. The Tenor Banjo has a shorter neck than the Plectrum Banjo, and is usually tuned differently.
A great deal of information about 4-string banjos, their uses and tunings may be found in the Four String Banjo FAQ which resides on Noel Induni's website.
These banjos are most often picked or strummed with a flat pick (a "plectrum"). These are the banjos usually hear in Dixieland bands, jazz bands, the Mummers Parade, and, of course, much Celtic music.
While there are many overlaps of style and music between tenor banjos and plectrum banjos, the former are typically used to provide rhythm, while the latter are often used for solo playing, as in this recording by the Banjo Kings of
You Are My Sunshine. Melodies may also be played in a chord style, heard here in The Banjo Kings' recording of
Jeanie with the Light Brown Hair. More of these soundbytes can be heard at tunes.com, and recordings can be purchased from that site.