You need to pass those tests to get an international certificate stating you general or/ and specific language knowledge. You can use the certificate to get into colleges and universities around the world, find a job abroad, retrieve lost citizenship and immigrate.
Special exams hold by different organizations. In USA it is Educational Testing Services (ETS) that developed well-known TOEFL test. In UK there are several boards that can certify you. All the tests united into 9-level system which makes it possible to compare them to each other. You need to double check if the chosen university accepts results of the test you’re going to pass. Another kind of tests is specialized: they are interesting for future lawyers, medical workers and MBA.
Usual exam itself usually consists of grammar, comprehension, writing and speaking. Age Restrictions Tests of general knowledge of language don’t have age restrictions. Some special exams do have, like TEFL (Teaching English as Foreign Language) – you can get it since 18 y.o. On other hand, it’s not very bright idea to teach before that age. Academic tests like GRE and GMAT you can pass since 25 y.o. Some children tests allow only kids younger 16 y.o. participate.
Standard cost if $30-150. But there are more expensive ones. And remember that is only the test payment, you need to prepare for it separately, by yourself of pay to some school. If you fail test, you need to pay again for another try.