The key to understanding these three phrases is “else”. Else means something or someone besides or in addition to the first option.
Anyone else meaning
We use “anyone else” when we need one or more people/things in addition to or instead of the first option.
Teacher: Who knows what the capital of Brazil is?
John: Sao Paulo
Teacher: No, that is incorrect, anyone else?
We can use anyone in negative and interrogative(questions) sentences.
Is anyone else or are anyone else?
“Is anyone else” is correct. Anyone acts as a singular word.
Is anyone else here from India? (this implies that I am from India)
Anyone else or anybody else?
Anyone else and anybody else have the same meaning.
Anyone else is probably more common in writing.
Someone else meaning
We use “Someone else” when we need another person that is not the first option.
I don’t want to do this. Can someone else do it?
We can use someone in positive and interrogative(questions) sentences.
“Is there someone else?” meaning
“Is there someone else?” is a question that people often ask when a romantic relationship is ending.
We use the question “Is there someone else?” when we want to know if the other person is having an affair(dating another person at the same time).
Everyone else meaning
We use “everyone else” when we want all the other people to do the same thing as the first person or something else.
Teacher: Ok, John, you sit on the floor.
……..Now, everyone else look at John.
In this example “everyone else” includes all the other children in the class.
Everyone else or Everybody else?
Everyone else and Everybody else mean the same thing. Everyone is a bit more formal and more common in writing.