Neither the term "rare earth", nor "family name", apply to phosphorus.
The rare earths refer specifically to a series of about 17 elements, predominately in the lanthanide series, which are considered rare because they tend not to appear in significant amounts in any one location or ore. Phosphorus is not located anywhere near the lanthanide series on the periodic table, and is in fact abundant throughout the earth.
The term "family name" is just another word for each column, or group, of the periodic table. The noble gases and halogens are one example. The group in which phosphorus is located, Group 15 (or occasionally Group 5A), are called the pnictogens, or the nitrogen family. Phosphorus is otherwise generally referred to as a nonmetal, rather than referring specifically to its family name.
Phosphorus is the name of a chemical element which is located in the nonmetals group 15 of the periodic table. The florid property of phosphorus was influential in its name selecting process, hence, the greek word "phosphoros", whose meaning is morning star, seemed the best pick at that time. The person that discovered phosphorus was the alchemist Hennig Brand, who isolated phosphorus from urine.
Phosphorus has three allotropes: white, red and black or violet.
The white allotropic form can spontaneously burn and it is toxic , hence it needs to be stored under water. The conversion between the white form and red form can be made exposing the white form to sunlight. The red form does not spontaneously burn but its handling needs prudence because frictional heating turns back the red form into the white form. The red form can be found in safety matches and fireworks.
Because the third allotropic form, black phosphorus, is not at all reactive, its commercial use is meaningless.
Phosphorus is needed to maintain the acid-base levels in the human body and it is also crucial, after calcium, in the composition of bones (85% in bones) and teeth. High levels of phosphorus in the blood can cause several disorders such as: extra levels of parathyroid hormone released, bone and heart problems, calcifications of tissues, itching and bone pains.