The details of work permits will vary, perhaps greatly, from country to country, but basically, a work permit is “permission” for an employer to employ you, a foreigner. In general, employers are required to find a citizen of their country (or in many parts of the EU, a citizen of any EU country) to fill a position before hiring a non-citizen, and if they can’t or don’t, they generally need to show why you, a foreigner, are better qualified than a local. Fortunately for native English speakers, one “requirement” for teaching in many schools is merely being a native English speaker, but your BA and TEFL certificate help show that you are qualified beyond having the good fortune to be born in an English-speaking country. Employers and expats talk about work permits in many ways, but generally a work permit is not something you “get” for yourself and then go work wherever you’d like – it is very often connected to and valid only for a particular employer.
Connected to a work permit is residence, which more or less official permission to reside in the county. The requirements again vary from country to country, but often include the offer of a job that pays a living wage, proof that you have no criminal record in your home country, a signed lease, and proof of health insurance. Sometimes there is a catch for teachers who go overseas to do their course: you are supposed to apply for the work permit at a consulate in your own country – but if you haven’t completed a TEFL course or traveled recently to the country where you’d like to work, you are unlikely to have a job offer or a signed lease.
There are often ways around this, such as applying for your permits at a consulate outside the country you intend to work in but in a neighboring country (e.g. if your job is in Hungary you go to the Hungarian consulate in Austria). If you don’t have a signed lease, you may be able to get a written statement from a local with housing that they will accommodate you.
It’s rarely simple, but be warned that the requirements for working or even traveling in the US (and other English-speaking countries) are probably some of the strictest and most rigid in the world, so be careful before you publicly complain about regulations in another country.