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Two-item reference, and the verb agreement each one forces.
Why you specifically

Both, either và neither đều liên quan đến hai đối tượng, nhưng cách dùng không giống nhau. Both nghĩa là “cả hai”, nên động từ thường chia số nhiều: Both of my friends are here. Either nghĩa là “một trong hai”, nên thường đi với động từ số ít: Either of the answers is correct. Neither nghĩa là “không cái nào trong hai”, và cũng thường đi với động từ số ít: Neither of the options is good. Lỗi rất hay gặp của người Việt là nhìn thấy danh từ phía sau là số nhiều rồi chia động từ số nhiều, ví dụ: Either of the answers are..., Neither of the rooms are.... Trong tiếng Anh chuẩn, động từ phải theo ý nghĩa “một” hoặc “không cái nào”, không theo danh từ số nhiều ở phía sau. Khi both nối hai chủ ngữ bằng and, động từ dùng số nhiều: Both Anna and Tom are late. Neither cũng có thể đi với danh từ số ít: Neither child was hungry. Khi làm bài, hãy xác định ý nghĩa trước: “cả hai”, “một trong hai”, hay “không cái nào trong hai”, rồi chọn động từ theo ý nghĩa đó.
The rule
Both, either, and neither all refer to two things, but they are not used the same way. Both means the two together, so it usually matches a plural verb: Both of my friends are here. Either means one of two, so it usually takes a singular verb: Either of the answers is correct. Neither means not one and not the other, and it also usually takes a singular verb: Neither of the options is good. A very common mistake is to make the verb plural after either and neither because the noun after of is plural. In standard English, the verb agrees with the idea of one, not with the plural noun inside the phrase. When both joins two subjects with and, the verb is plural: Both Anna and Tom are late. The noun after neither can be singular: Neither child was hungry. With both, either, and neither, always ask whether the meaning is two together, one of two, or not one of two, then choose the verb form from that meaning, not from the noun that follows.
The trap

Both of the books is on the table.
Both of the books are on the table.
Both refers to two things, so the verb is plural. We usually say both of + plural noun or both + plural noun.
Either of the answers are correct.
Either of the answers is correct.
Either means one of two, so we use a singular verb in this structure.
Neither of my parents are at home.
Neither of my parents is at home.
Neither means not one of two, so it takes a singular verb here.
Both Anna and Tom was late.
Both Anna and Tom were late.
When both and and join two subjects, the verb is plural.
Prove it
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