There are many different types of banjos, but the most popular, or at least well-known is the 5-string banjo, which is played most commonly in American Folk, Country, Western, and Bluegrass music. The 5-string is usually played by individually plucking the strings in a fast, rhythmic, "rolling" motion using individual finger picks.
Other common varieties are the Tenor Banjo, which has only 4 strings and is tuned in 5ths like a mandolin or a cello. Similar to the Tenor banjo is the Plectrum Banjo, which also has four strings but is tuned slightly differently.
The Tenor and Plectrum Banjo are more commonly found in early Jazz music, and were usually just strummed rather than picked, much like a rhythm guitar. Plectrum and tenor banjoes were quite common in popular music during the later half of the 19th century until around the mid-1930's. They are now considered more of a specialty or novelty instrument and are not often seen outside of nostalgic jazz bands.
There is also the Banjo Ukulele, which is a very small banjo that is tuned like a ukulele, the long-neck banjo, which is similar to the regular 5-string, but has a much longer neck (big surprise), and the 6-string banjo, which is basically the neck of a 6-string guitar and the drumhead body of a banjo.
Because of the banjo's common appearance in folk music (mostly the 5-string banjo), especially in bluegrass and music which originated in mostly rural or undeveloped parts of the united states, especially the more mountainous parts of the southeastern United States, it has become a symbol of the (arguably well-deserved) stereotypes associated with those parts of the country, such as being unhygenic, backwards, ignorant, simple, inbred, incestuous, and bigoted. The banjo has long been associated with hillbillies. The infamous 1972 motion picture "Deliverance" played a large part in galvinizing the banjo's association with the uneducated, inbred, redneck hillbilly with a scene featuring a boy (or growth-stunted young man) who appears to possibly be the product of incestuous union. The "boy" does not appear to be able to speak or communicate, in fact hardly seems capable of having facial expressions, but is an absolute savant at playing the 5-string banjo.
Because the many variants of the Banjo have been largely present in antiquated or venerated styles of music, mainly those of American origin, the banjo is also associated with being old-fashioned.
For I come from Alabama with a banjo on my knee."
"Someone's in the kitchen with Dinah strummin' on the old Banjo."
"That guy's so inbred, all he can do is play the banjo."