Visas for Teaching in China
For teaching in China, ideally you want a work visa, often known as a Z visa - Z being the first letter of the Chinese word for work. To get one of these, you need to apply to a Chinese embassy with documents supplied by your employer. Only certain schools, licensed to employ foreigners, can supply these.
Business (F) visas are sometimes used by teachers, usually freelancers. Strictly speaking it is illegal to work on a business visa - they are for people cooperating with Chinese companies, not working for them. However, it's very common to find people working on business visas, especially in the larger cities.
Tourist (L) visas are the easiest to get hold of. They are usually only valid for 30 days, but can be extended. Teaching in China is illegal on a tourist visa. Employers may ask you to come in and work on a tourist visa, and claim they will help you change it to a work or business visa. See below for more info.
If your employer supplies all the necessary documents for a work visa, you can be sure they are licensed to employ foreigners. Obviously, this does not guarantee you a perfect experience, but its a step in the right direction.
Teaching on China on a business or tourist visa
When you are asked to start on a business or tourist visa, with the promise of changing it to a work visa later, be on your guard. The only reason for an employer to do this is if they are under time pressure and don't have time to arrange the documents, or they aren't actually allowed to employ foreigners. If its a matter of not having enough time, and you're confident the school can get you the work visa, I'd be happy enough. Otherwise, I would be very wary and at the least want to talk to a former teacher who'd gone through the same process.
If you wind up teaching on a tourist or business visa, and the school is unable to get you a work visa, it's not a complete disaster - I've seen people stay in China for a year and a half renewing business visas every three months - apart from running out of pages in their passport they didn't have any problems. However, it does put you on legally shaky ground and although the school will likely bear the brunt of any official displeasure, it's not a situation I would let myself be in for more than a month or two. Apart from the fact that it indicates your school isn't as well-established and legal as you might wish, it can put you in a very difficult position if there is any problem between the school and the PSB.