Data modelling concepts

Generally a many-to-many relationship is done with a junction table, like so:

Student (id, name, date of birth, department) 

Course (id, name)

Student_Course (student_id, course_id)

You can either create an artificial key on the Student_Course table or create a primary key or unique index with both fields. student_id is a foreign key to student.id and course_id is a foreign key to course.id.

Note that the junction table need not be exactly two fields. You can include fields such as date_added or date_dropped so you could see when students have added and dropped courses.

There are very few scenarios in which you would probably benefit from your second design:

One is if there is a sub-typing scenario where only certain a certain sub-type had the relationship (and the super-type doesn't) and the sub-type has other additional fields, besides the foreign key.

Another might be where the existence and value of the foreign key needs to be controlled for security reasons at a different level of granularity than the rest of the entity containing the foreign key. Most DBMSs can handle security per table. Most don't handle security per column.