Choosing the right strings used to be important, but with modern production methods there's not that much quality difference between the various brands.
Gut and nylon strings
These are hardly ever used for Celtic music (unless you play a zither banjo and want to go the really traditional way), but for the sake of completeness:Just like metal strings, a gut/nylon string set has plain and wound strings.
The plain strings are nearly always made from nylon these days. Genuine gut strings do have a lovely and very unique tone, but nylon strings have more volume and tonal flexibility, are much easier to handle and cost much less. Recently there have been attempts to make synthetic gut strings from sepcial materials developed to emulate the tone, look and feel of gut as closely as possible. Whether these will catch on remains to be seen.
The wound strings are usually made from a core of silk threads or silk-like synthetic threads wound with a thread made from some bronze or silver alloy.