What was the belief held during the cold war regarding neighboring countries of communist countries all falling victim to communism known as?

It was called the Domino Theory.

This theory said, if a country becomes communist all its neighbors will eventually fall to communism like dominoes.

The reality is, countries don't become communist unless they're forced to. People like owning things. They like being able to go to the store and buy bananas if they want. And it's hard to do those things in a country where the state owns everything and foods are rationed.