Do universities employ teachers?
Yes, research universities hire full-time faculty whose primary job is teaching.  At least in the US, these faculty typically have different titles from tenure-track faculty, such as instructor, lecturer, senior lecturer, and teaching (assistant/associate/full) professor.  For example, my department employs 12 lecturers and teaching [rank] professors; my wife is one of 13 lecturers in another department.  We work at a large R1 state university; both our departments also have lots of "standard" tenure-track faculty.

Teaching faculty in my department are primarily responsible for the large lower-division undergraduate courses.  To first approximation, each such class is taught and maintained by the same teaching faculty member every semester.  In contrast, tenure-track faculty like me normally bounce between a few different courses, including upper-division requirements, electives, and graduate classes.  Teaching faculty in my department typically teach 3-5 times as many students per semester as tenure-track faculty, mostly because our intro classes are huge.  Most teaching faculty also have teaching-related service responsibilities, like serving on curriculum revision, grievance, and appeal committees.

At most R1 universities, including mine, teaching faculty are ineligible for tenure. But there are exceptions, like Berkeley's "Lecturers with Security of Employment" or Duke's "Professors of the Practice".