I taught English as a foreign language in France, the Czech Republic and Scotland. There are a number of qualifications available for doing so, but the most widely accepted are the TESOL as mentioned above and the CELTA, which is a Cambridge University moderated qualification. I recommend the CELTA, and if you browse for teaching jobs online (try tefl.com) you will find that most require applicants to have this certificate.
The CELTA can be obtained before you go; my husband studied for his in NY before traveling to Indonesia to teach. You can also study for the certificate in Europe; I studied for mine at a language school in Paris then searched for work from there. There are pros and cons either way. If you find a school near you which offers the CELTA training course it might turn out cheaper to do it that way. However, in my day it cost around 1,000 pounds to do the course, and it cost less to do it in Paris. I just had to find somewhere to stay for the duration. Countries like Spain and Portugal seemed to offer the cheapest CELTA courses, but some googling will tell you where the nearest courses are to you, and how much they cost in places in Europe. If you do decide to do it in Europe, the school where you study for it will usually consider hiring you as a teacher as soon as you qualify, which is a real bonus as it takes away the stress of job hunting.
If you want to teach in mainstream schools in Europe, you will need more than a TEFL diploma to do so (eg a post-grad in teaching). Once you have been teaching for two years with a CELTA you can apply to do a DELTA, which is the next level up and pretty much equivalent to a post-grad. This would allow you to teach in International Schools in Europe, or just move up the ladder in private language schools.
TEFL teaching is notoriously badly paid. If you want an experience and something good to put on your CV I would highly recommend it, but you couldn’t raise a family on it unless you had been working for a few years and had managed to secure a senior teaching post.
If you’re American you will have difficulty teaching (legally) within the EU. Countries that are relatively new to the EU (like Czech Rep) offer posts to Americans. Asia is an exciting option for Americans, and you could potentially earn a lot more teaching there than in Europe. You may find though, that jobs in Asia are more often than not based in public schools or universities. Some find this kind of teaching less interesting than in private language schools where you would deal with a wider variety of clients, from children to business folk.
For more info try tefl.com as mentioned, or Dave’s ESL Cafe at eslcafe.com.
Hope this helps!