You can find an illuminating list of banjo tunings at the Deering Banjo Company’s site: https://www.deeringbanjos.com/pages/how-to-tune-a-banjo. The article covers the five most important of the many tunings. Take a look! The Deering site provides the visitor lots of banjo information, whether she/he buys a banjo or not.
G tuning—D, B, G, D, G—dominates, but the C tuning—D, B, G, C, G—is very popular, and ranks as a second standard. (The Deering article gives the order of the strings 5th to 1st, e.g., G, D, G, B, D)
There are two reasons to choose a nonstandard tuning. The first is simply to make it easy (or even possible) to play in a particular key. That’s clearly the motive when a player just tunes the 5th string down to F# when playing in D. If your voice (or your band) can’t adjust to the standard keys, you have to do something! Of course, the capo-ble banjo solves this problem with a capo.
The other reason to use a nonstandard tuning is to adjust to a particular song or style. (See Pete Seeger How to Play the 5-String Banjo for some examples.) A capo-ble banjo can be retuned for this reason as easily as a standard banjo.
D tuning pegs are a modern twist on this. These allow the banjoist to switch the 2nd and 3rd strings from G tuning to D tuning and back while playing. D pegs are expensive and a little finicky, but full of possibilities for the professional performer.