Both は2人の人や物について話すときに使います。

Both cats and dogs are pet animals.

決定詞 (the, my, this など) を伴う名詞の前では both and both of が使えます。

Both of my parents are teachers.

This are two of teacher. OR Both my parents are teachers.
You can take both of the shirts.あなたは両方のシャツを取ることができます。 OR あなたは両方のシャツを取ることができます.
これらの答えは両方とも正しいです. OR Both these answers are correct.

Note that is often dropped after both.

She has eaten both chops.彼女は両方のチョップを食べました。 OR She has eaten both the chops. OR She has eaten both of the chops.

Before personal pronouns we use both of.

Both of them have agreed to come.Or She has eaten both of the chops.

Both of them have agreed to come. (NOT Both them have agreed to come.)
She has invited both of you.彼女はあなた方両方を招待しました。 (NOT She has invited both you.)

bothの後には代名詞の目的形（us, them, youなど）が続くことに注意。

the not used before both

We do not put the before both.

Both the boys were given prizes.The two of students are used for both. (NOT The both boys were given prizes.)

否定構文

否定構文では、not both notではなく、nitherを用います。 (NOT Both of them did not come.)

Both with verbs

Note the special structure in which both goes with the verb.

They can both swim.（彼らは両方とも泳ぐことができる）。 (=どちらも泳げる。)
They have both come. (=両者とも来たことがある。)
They both liked the movie. (= Both of them liked the movie.)

Both…and

Note that words of the same part of speech follow both and and.

She is both beautiful and intelligent.この構造では、同じ品詞の単語がbothとandの後に続くことに注意しましょう。 (ここではbothとandの後に形容詞が続いています。)
She both sings and acts. (動詞)
私はジョンとピーターの両方を招待しました。 (名詞)