In linguistics, the term Language Register is used to talk about the type of language a person might use in a certain social context, in other words, how formally they will speak.
No register is right or wrong per se. What determines this is how appropriate the language used is in a certain situation.
Suppose as a teacher a student asks about the meaning of the word cop. It is simple to explain that the meaning is police officer but it is used in certain situations or as part of a certain register. An individual might refer to a police officer whilst giving a formal statement to the police: Three police officers arrived in a car ten minutes later. However, later when speaking to their friends, the same individual might say: Three cops turned up in their car ten minutes later.
Thus, register depends on:
As a general TEFL teacher the most common reason register comes up is when dealing with what language is suitable in formal vs informal situations.