There are plenty of commercial recruitment agencies specialising in the English language market – working for a school to help find teachers or helping a teacher find a job. The agencies are normally paid by the employer (the school), though make sure that you check before you sign up! Agencies normally cover a particular area or country – and will help you put together a good CV, send you new job opportunities and even advice on relocating, accommodation and visas.
Almost all the jobs for ELT or ESL teachers are with English language schools. The majority of all schools are run on an individual basis, with the director of studies also the person who interviews and appoints new teachers. But there are some major chains of schools who have their own internal recruitment agencies who can help you find a job within the schools in their group. Once you have a job in the group, the internal job-agency often works to help find a job in another school within the group or in a different country.
The biggest ELT employer in the world is the British Council, which has its own recruitment service – see here for more details on the British Council. As a good starting point, try the commercial agencies and major international school groups listed here.