The Academic Year is when a school or learning institute is open for business.
In many countries this is from September/October to the following May/June.
In some countries in the Southern Hemisphere, the academic year runs from February/March to November/December. Obviously then an academic year for most schools lasts about 10 months or so (including holidays).
Most schools divide the academic year into either 2 terms (or semesters) or 3 terms (in the UK these are the Autumn, Spring & Summer term; elsewhere they are usually just numbered).
Some schools, in particular those involved with teaching students from abroad, do not have an academic year as such and remain open all year round. These types of schools run short courses of anything from a week to a few months.
Most contracts for TEFL teachers will be for a single academic year; occasionally it could be longer and very occasionally it could be shorter.
For details of when schools are operational in different countries, see the Country Guides for TEFL Teachers.