Refraction is the bending of light as it passes from one medium into another, and is the underlying mechanism by which a Lens is able to focus light.
Index of Refraction
The Refractive Index (or Index of Refraction) of a material is the degree to which the speed of light is reduced in that material, relative to vacuum. The amount (and direction) of the bending at the interface is dictated by the relative values of the Refractive Indices.
The Refractive Index of air is close to 1 ; that of glass is around 1.5. The higher the Refractive Index, the thinner a lens needs to be in order to achieve a given Focal Length. Higher refractive index is not always better however, the Refractive Index typically varies depending on the wavelength of light, which means that different colors of the spectrum will bend to different degrees, leading to Chromatic Aberration.
There is also an element of reflection at the boundary between two media. (Think of how a window into a dark room acts as a mirror on a bright day.) This can be reduced using an Anti Reflective Coating - a thin film of a material with a Refractive Index of an intermediate value.
The term Refraction (and also Refractrometry) is also used as the study of the Refractive state of the eye, by which an optometrist determines your ideal correction for "best corrected visual acuity". You can do your own trial refraction at home with a trial lens kit.