Graded language is classroom language that is adapted to the level of the learners in some way. Many course books use graded language. It can be compared with authentic language, which is not changed in any way.
An intermediate level class are using graded readers, reading works of literature but written in simplified English.
In the classroom
Most learners using course books are exposed to graded language, so it is useful to try to include authentic language and highlight the differences between them. For example, the teacher can invite a native speaker into the class for a listening exercise which mirrors one the learners have done from a course book