Radius refers to the curvature of a fret or fingerboard. The fret curve can be imagined as part of a circle, and the radius describes the distance of each part of the fret from an imaginary center.
The larger the radius, the larger the circle, and the flatter the fret surface will be.
In general, the smaller (i.e. more curved) radius is more comfortable for playing chords (particularly barre chords).
By contrast, a larger, flatter radius is better for low action, single-note playing and bending. A flatter radius has the advantage of making a lower action possible.
This is because when you bend a string on a lower part of the fretboard such as your first or second string, you are bending it towards the middle of the fretboard, which is higher than the edges. If the radius here is too large, the notes will tend to “fret out” – i.e. buzz against the higher parts of the fret, killing sustain.
The best answer to this type of problem lies in the implementation of a so-called compound radius.