meaning - "A number of students" vs. "the number of students"

From the grammatical view both are correct, but please explain the difference in meaning:

1. The number of students in the class is fifteen.


2. A number of students were late for class.

Answer

The number of students in the class is fifteen.

The verb is singular because it refers to 'the number'. The subject of this sentence is 'the number'. 'Of students' is a modifier of 'the number'

'A number of' means several, some.

Several students were late for class.

Some students were late for class.

A number of students were late for class.

The subject of these sentences is 'students', and 'some', 'several', 'a number of' are modifiers of 'students'. The verb agrees with the subject.