Okay kiddo, when you hear someone say the correct name in botany, they are talking about the proper name for a plant. You know how you have a name to identify you and tell people who you are? Plants also have names called scientific names.
This scientific name has two parts, the first part is called the genus and the second part is called the species. Think of the genus like your last name, and think of the species like your first name.
For example, if we talk about a sunflower, the scientific name would be Helianthus annuus. Helianthus is the genus name and annuus is the species name. This helps people all around the world know exactly what plant you are talking about, no matter what language they speak.
So, the correct name in botany is really just using the right scientific name for the plant you are talking about. It's kind of like calling your friends by their real name, instead of a nickname or something else.