In much of Asia it is possible to find work in language centres all year round. This is because of the high demand for English tuition from native speakers and also because there are often new course modules and classes starting every couple of weeks.
However, if you want to teach at a state school or International School then there will be term dates that dictate exactly when you will start. It is sometimes possible to start halfway through the academic year but most schools much prefer to hire teachers to start the new school year and sign the contract for either one or two academic years.
The most common time for the academic year to begin is late August or early September. However, do bear in mind that the recruitment process will have begun several weeks prior to this, and in some cases several months prior. Public schools in South Korea usually recruit between April and August to start at the very beginning of September or last week of August. The JET program in Japan usually recruits in December for a September start the following year. The NET scheme in Hong Kong requires candidates to apply by the end of January for an August start.
In Muslim-majority countries such as Malaysia and Indonesia, no hiring will usually take place during Ramadan. This month-long religious period changes dates every year so do a quick Google search to ensure that you avoid that time.
In Europe, the most vacancies tend to be at the end of summer (i.e. September) but managers will begin the recruitment process several weeks before this. Given that a considerable number of vacancies are not advertised, it is hugely beneficial if you are already in your European country of choice, with all the necessary documents such as your TEFL certificate and a pile of passport photographs, by mid-August.
Native English speaker teacher programs managed by governments in France and Spain start their recruitment drive much earlier. The start dates for these tends to be at the beginning of September but the recruitment will have been done by March at the latest. Short-term summer schools in Europe tend to recruit from late November onwards (but mostly in March/April) for contracts lasting for several weeks during the May-August period.
There is sufficient demand for English tuition in Eastern European countries for recruitment to go on much of the year.
In the Middle East, language centres recruit all year round, but state schools and private schools usually recruit between March and July for the beginning of the new school term in August or September.
In Latin America, speak hiring months are December/January and July/August, although for the majority of these jobs you will need to be already in the country as very few recruit from overseas. In countries where there is high demand for English tuition, such as Chile and Mexico, it should be straightforward to find work most of the year.