Now the teacher plays the tapes or reads the text again, pausing after every sentence to check the students understand.
Is ‘understand’ a noun in the sentence above?
No, it's not a noun.
The normal wording of that sentence would be, "... pausing after every sentence to check that the students understand." Sometimes people leave out the word "that" from dependent clauses like this.
"The teacher knew that her students understood the assignment."
is equivalent to
"The teacher knew her students understood the assignment."
In the particular case quoted in your question, I don't think it sounds exactly proper, although it's probably acceptable. But even without the word "that", "understand" is still a verb with "students" as its subject.
The word “understand” is a verb, referring to the action of comprehending. As such, the sentence as is is incorrect. However, it can be easily rectified by adding the letters “ing” to the end. The word “understanding” in this context would indeed be a noun, referring to the state of the students’ comprehension.