Fretboard radius pertains to the curvature of a guitar neck, left to right when looking down the neck, headstock to body.
The easiest way to understand this concept is to relate fretboard radius to the two most popular guitar types, Fenders and Gibsons. Fender guitars are known for having a lower radius (more curve) and Gibsons typically have a higher radius (flatter neck).
With this in mind, one can relate the different types of music and playing styles associated with either distinct brand and therefore make a pretty good assumption about whether a higher radius (flatter neck) or a lower radius (more curvature) would better accommodate a personal playing style.
Pickguardian provides a free, printable gauge guide that can be used to estimate fretboard radius. Keep in mind, the lower the number the more curvature and the higher the number the flatter the neck.