Learn Russian: Соседка жалуется, что ей некого позвать на вечеринку. - The neighbor complains that she has no one to invite to the party.

This is the core structure for predicative negative pronouns like некого (there is no one) or нечего (there is nothing). In Russian, you cannot translate 'she has no one to invite' word-for-word. Instead, the structure is literally 'to her there is no one to invite', which requires the logical subject to be in the Dative case (ей), followed by the negative pronoun and an infinitive (позвать).

Why is it некого instead of некому or некем?

The case of the negative pronoun depends on its role with the infinitive verb. Here, the verb is позвать (to invite). Since you invite a direct object, we need the Accusative case. Because we are talking about people (animate), the Accusative form is некого.

Why isn't it никого? What is the difference between некого and никого?

Никого is used in standard negative sentences with a conjugated verb and the particle не (for example: она никого не зовёт meaning 'she is not inviting anyone'). Некого is an independent predicative pronoun that already contains the idea of 'there is no one to do something' and is always used with an infinitive, without an extra не.

Why do we say на вечеринку instead of в вечеринку?