Different schools and language centres have different policies on this issue. Sometimes you will see a post specifically targeting a couple. In these circumstances, the thinking would appear to be that if a teacher has a close friend or partner to support them as they make the transition from one country to another then the whole settling-in process will be a great deal smoother.
Some schools prefer not to hire teaching couples for several reasons. If one leaves, the other may follow soon after. If you are unmarried, the local culture may not be approving of you living together although this also depends on the context such as whether you are in a large city or a small town. If in doubt, speak to your potential Director of Studies or Manager at a suitable time prior to arranging accommodation.
If only one of you is a qualified ESL teacher then it is crucial to do some research on temporary residency permits. If you are married but one person plans not to work, then it is usually quite straightforward to apply for a spousal visa. Do bear in mind that this process may need to be started several weeks before you arrive and you must check the contract carefully with regard to whether or not permits or visas for your immediate family will be arranged and paid for by the company. It is not common, although some of the more lucrative positions in the Middle East include the cost of airfares for your family as part of the package. Most British Council postings include a substantial ‘settling-in allowance’.
If you are unmarried, and only one person is a teacher, it might be more difficult to benefit anywhere near as much as married couples. You should also check how long the accompanying partner is allowed to stay in the country on a simple tourist visa and whether or not it can be renewed. These rules and regulations change frequently so check on the embassy websites for the latest information and any downloadable forms to fill in.
If you are married and one partner is a citizen of the country you intend to teach in then the process of getting the necessary paperwork completed is usually much less complex. Again, this varies from country to country, but many nations give you the right of residence if you a married to a citizen of that country. Unfortunately others only give you this right after you have been married for over five years. Once again, these rules can change at short notice so be sure to ask for the latest advice from official embassy websites.