There are plenty of different members in the mandolin family, but one thing they all have in common is the need for a good set of strings to give you the best sound. By far the most common string configuration is four paired courses which, when using a great set of strings, gives the mandolin its characteristic tremolo capability. Just a reminder—if you're an advanced mandolinist playing six-course instruments or three-string courses, you'll need more than one set to fully string your mandolin.
Each set of strings can have a very different sound—this is perhaps even truer of mandolin strings than other stringed instruments. You've probably seen this firsthand the first time you re-stringed your mandolin with strings that turned out to be completely different from the original ones. Sound is a matter of taste, so it will probably take some experimentation for you to decide on your personal favorite strings.
Despite the variance of mandolin strings, there are a few general characteristics to inform your choice. The timbre of a string depends on what it's made of. Bronze strings, for example, are usually held to be the brightest-sounding. Phosphor-bronze is similar, but with a richer tone and lower brilliance. You'll get a heavier bass sound from materials like stainless steel, nickel and tin or nickel-plated steel. Apart from their sound, stainless steel strings are also notable for their corrosion resistance, which makes them a solid choice if you live in a humid area or find that your hands tend to sweat while you're playing.
The other main attribute of mandolin strings is their gauge. This refers to the thickness of the strings, with strings of different gauges vibrating at different tensions to produce the same pitch; the higher the number, the heavier the gauge. Lighter-gauge strings are more flexible, vibrating fast and producing brighter tones. Heavier gauges, on the other hand, deliver greater sustain. Avoid heavier string gauges if you are playing a mandolin with high tuning, though: they can wind up putting more tension on the instrument than it's designed to withstand.
Your ideal strings will be those with the perfect combination of material and gauge to give the exact sound you prefer. You may even decide to 'split' sets, taking certain strings from one set and some from another. If you want a string with a smoother feel on your fingers, look into coated or flat-wound strings. No matter what your standards are for sound and feel, there's a set of strings that will fit.