Focal length and radius of curvature are terms used when discussing two of the wave-like behaviors of light: reflection & refraction.
To some degree, all waves (including light) bounce off obstacles they come in contact with, and this "bouncing off" is referred to as reflection. Most commonly, we talk about seeing reflections in things like puddles of water, car windows, and mirrors.
Sometimes, waves are able to pass through obstacles they come in contact with, like panes of glass. When waves (including light) pass from one medium to another, "bending" occurs, and this is referred to as refraction.
Both reflection & refraction can occur in a myriad of situations, but often times the most interesting situations to study are those involving curved obstacles, like mirrors or lenses, because the resulting image differs from the image you started with! And it's when scientists deal with these curved obstacles that focal length and radius of curvature become relevant.
Imagine that the curved obstacle extended its edges to form a full circle. At some point within the full circle is its center-most-point (or center of curvature), and the distance from the center of curvature to the outer-most-point of the obstacle (or vertex) is referred to as the radius of curvature.
Halfway between the vertex and the center of curvature lies the focal point, and the distance between the focal point and the vertex is referred to as the focal length.
Ultimately, the following relationship exists:
(focal length) X 2 = (radius of curvature)
NOTE: Although wave-like behaviors of light are discussed above, light is really believed to abide by a "wave-particle duality."