Both se usa para hablar de dos personas o cosas.

Tanto los gatos como los perros son animales de compañía.

Ante un sustantivo con un determinante (the, my, this etc.), both y both of son ambos posibles.

Both of my parents are teachers. OR Both my parents are teachers.
You can take both of the shirts. OR You can take both the shirts.
Both of these answers are correct. OR Both these answers are correct.

Note that the is often dropped after both.

She has eaten both chops. OR She has eaten both the chops. OR She has eaten both of the chops.

Ante los pronombres personales se usa both of.

Both of them have agreed to come. (NO Both them have agreed to come.)
She has invited both of you. (NOT She has invited both you.)

Note that both of is followed by object forms of the pronoun (us, them, you etc.).

The not used before both

No ponemos the antes de both.

Both the boys were given prizes. (NOT The both boys were given prizes.)

Estructuras negativas

En estructuras negativas usamos neither, not both not.

Nither of them came. (NO Ambos no vinieron.)

Ambos con verbos

Nota la estructura especial en la que ambos va con el verbo.

Ambos pueden nadar. (= Los dos pueden nadar.)
Los dos han venido. (= Ambos han venido.)
A ambos les gustó la película. (= A ambos les gustó la película.)

Tanto…como

Nótese que en esta estructura las palabras de la misma parte de la oración siguen a tanto y y.

Ella es tanto hermosa como inteligente. (Aquí tanto y como van seguidos de adjetivos.)
Ella tanto canta como actúa. (Verbos)
He invitado tanto a Juan como a Pedro. (Sustantivos)