Lead was first added to glass on an industrial scale in the 1600's. It makes the glass denser and decreases the speed at which light travels through it. This causes refraction of the light. Lead Crystal has an index of refraction of n = 1.7, which means it reflects light greater than regular (soda-lime) glass and is therefore more brilliant. In addition the light is refracted into it's component wavelengths which is what causes rainbows.
An interesting note is that the term "crystal" actually refers to a particular property that pertains to solid matter when their atoms and sometimes molecules form an organized lattice structure in space. Ice for example is, at an atomic level, a crystal; when we look at it from an atomic level we see the atoms arranged in a structured pattern. Ironically glass does not have this crystalline property and therefore referring to crystal glassware as "crystal" is actually a misnomer.
Now a days true "Lead Glass" is rarely manufactured due to the health risks associated. Lead is extremely poisonous and hazardous; it can cause severe damage to various body organs and even death if exposed. The lead from lead glass containers contaminates stored liquids, therefore never ingest anything stored in lead glass! In place of lead barium oxide, zinc oxide or potassium oxide are used.
And so finally this brings us to the question I am sure you have all been asking yourselves: Why do crystal glasses sing, but soda-lime glasses do not?
The changes in properties of the glass when infused with lead are not only responsible for the physical differences between crystal and glass, but also the acoustic differences.