Wow, n-Dave gave an excellent recap of the situation... One sentence may have a typo, though...he may have meant to say "They can't convert an
L or an F visa."...
The types of visa are:
L: Tourist visa. Usually shorter-term. Can only stay in hotels. Illegal to work.
F: Business visa. Usually medium- to long-term. Can only stay in hotels or officially-designated housing. Can work for a foreign company, but not a Chinese one. (Note that nearly all "foreign-owned" schools are in fact registered as Chinese companies...)
Z: Work visa. Usually long-term. Permits one to enter China, then rent an apartment and start a job long enough to convert to a Residence Permit.
Residence Permit: Usually one year in duration; may be an academic year rather than a calendar year- this is tied to your contract. This permit conveys full legal permission to live and work in China. If you don't have this, and are not in process of getting one, you are an illegal migrant worker in an authoritarian state. Any school that tells you differently is lying to you for their own enrichment.
There are other visas for particular classes of people (students, diplomats, journalists, airline workers, etc.) but these usually don't interest prospective teachers.
Don't feel bad if you're confused. WE'RE confused. Many of the schools are confused, especially newer ones just beginning to work with foreign teachers. A surprising number of the very officials and police officers who supposedly administer these visas are confused, too. Different provinces, sometimes even different cities, can have different rules regarding visas. And beyond that, schools with good connections in the right Bureaus can quite often easily do things that the letter of the law says are impossible.
So, welcome to Club Confusion...
One point in particular causes massive confusion. Dave said it, but I'll say it again:
A visa- ANY visa- only gives you permission to enter the country, and carries a requirement to exit China before it expires.
Once you have a Residence Permit, visas are irrelevant and not needed. Your expiry date is now the date of the Permit, not the Z visa. The only time you may ever need a visa, once you have the Permit, is if you leave mainland China and then wish to re-enter. Most Permits today automatically come with re-entry privileges; if not, these are easy and inexpensive to obtain. If you ever leave China temporarily, even to go to Hong Kong or Macao, be sure you have re-entry privileges before you leave.
One last point: Yes, many teachers do stay and work here illegally on F and even L visas. A fair number of them get away with it. Many of them, and the schools they work for, will tell you it's fine. Please don't let this fool you:
YOU ARE RUNNING A VERY REAL RISK OF SUBSTANTIAL PENALTY, DETENTION, DEPORTATION, AND EVEN JAIL TIME IF YOU DO THIS. This course is absolutely NOT recommended.