Short answers, pulled from the story.
A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens is a single piece of transparent material, while a compound lens combines several simple lenses arranged along a common axis. Lenses are made from materials such as glass or plastic and are ground, polished, or molded to shape.
The word lens comes from lens, the Latin name of the lentil, because a double-convex lens is lentil-shaped. The lentil also gives its name to a geometric figure.
Lenses have been used as burning glasses for at least 2400 years. The oldest certain reference is in Aristophanes's play The Clouds, from 424 BCE.
Spectacles were invented in Northern Italy in the second half of the 13th century as an improvement on the reading stones of the high medieval period. This began the optical industry of grinding and polishing lenses, first in Venice and Florence in the late 13th century.
Chester Moore Hall invented the compound achromatic lens in England in 1733. The same invention was also claimed by fellow Englishman John Dollond in a 1758 patent.
A Fresnel lens has its optical surface broken into narrow concentric rings, allowing it to be much thinner and lighter while using less material. It was first fully implemented in a lighthouse in 1823.
A converging or positive lens, such as a biconvex lens in a lower-index medium, focuses a collimated beam to a spot behind the lens. A diverging or negative lens, such as a biconcave lens, spreads a collimated beam and has a negative focal length.
The main aberrations are spherical aberration, coma, and chromatic aberration. Other kinds include field curvature, barrel and pincushion distortion, and astigmatism, and even a flawless lens is limited by diffraction through its finite aperture.