In my experience, very few 4-year ("university") institutions in the United States still offer tenured positions to faculty without terminal degrees in their field (except in very unusual circumstances). In mathematics, the terminal degree is a PhD, and you will find that virtually all schools require a PhD in their job advertisements for tenure-track faculty. Sometimes an EdD is acceptable or even desired for teacher-preparation positions.
So, if you want to go into university teaching with only a master's you need to be aware first and foremost that you are not on the "usual track".
History
In the past, it was not uncommon for non-research-intensive institutions, such as regional comprehensive universities, to hire permanent faculty with a master's degree and give them tenure. This was partially due to a focus on teaching, rather than research, and partially due to difficulty attracting candidates to more remote, rural locations where these institutions are often located. Some institutions still have older professors who were tenured under this system, perhaps hired in the 1970s.
The historical factors are no longer as pressing. Even at non-research-intensive schools, there is more pressure for research than in the past. And there are so many new PhD graduates each year that the job market for tenure-track jobs is already saturated with them, even at non-prestigious schools. Institutions also have some pressure from accreditors to have courses taught by faculty with terminal degrees.
Options for master's degree holders
There are still two realistic options for finding mathematics faculty positions with only a master's degree:
Some institutions, particularly larger ones, hire a number of full-time "instructors". These positions are usually not tenured, but they are often full-time (w/ benefits) and on renewable contracts.
Community-college still often hire faculty with master's degrees, in my experience. These institutions often have their own version of "tenure" (which may go by another name).
To maximize your chance of getting a position at these institutions, you need to focus on building a strong vita in teaching. Community colleges, in particular, are often looking for a very different skill set than 4-year universities.
There are also part-time ("adjunct") jobs, where you are paid by the course. These typically don't have benefits, often do have lower salary, and according to many accounts they are a very difficult way to make a living.
Things to consider
In the job market, you have to watch out for yourself and be your own advocate. The director mentioned in the question may genuinely believe he is offering you a good position, but he probably has other pressures as well, such as finding instructors for all the courses. So be careful to do your own research into the department. If the department has faculty without a PhD teaching real analysis, that is not a deal-breaker, but it is a red flag that needs to be investigated.
You always need to pay close attention to the salary and job security in these positions, which vary greatly between institutions. For example, some institutions have strong unions, which lead to job security that is comparable to tenure. Others have essentially no job security. Salary also varies widely (it won't ever be extremely high, but it could moderate or extremely low). Beyond your starting salary and benefits, you also want to investigate whether the institution has a regular policy for raises for non-tenure-track faculty, if you are planning to keep the job for a while.
If you are also considering PhD studies, you should be aware that teaching for several years with a master's is not likely to help your odds for admission to a PhD program later. The job options for a PhD are also very different, which is not to say that they are always better. So you should take time to decide what track is best for you, before making a commitment to either one.
