There are three official French proficiency tests, although there are not three different tests. They simply correspond to different proficiency levels:
The tests are administered by the Centre International d'Etude Pédagogiques.
DILF/DELF/DALF are the French equivalent to IELTS or TOEFL, but this is where the comparison ends. With these French language tests you wont get a score and each test indicates your proficiency level. So the most advanced the diplôme held, the higher your proficiency. This means (as you may have noticed!) that, unlike IELTS or TOEFL which require only one test, you might have to sit several tests. The three tests are split across seven levels, all delivered by Ministère de lEducation Nationale (and related organisations abroad like Campus France or cultural institutes). The seven levels are set within the European Framework of References for Languages (CEFRL).
If you are applying for postgraduate study in higher education institutions in France (e.g. universities, Grandes Ecoles) or in other French-speaking countries, youre most likely to require a DELF 5 as a minimum, but some institutions may require a level as high as a DALF 6.
Each of these levels will be an indication of your level in four linguistic proficiencies: reading, writing, listening, and speaking. The tests are of different length depending on the level, so the DELF exams will be between 80 and 150mins while the DALF 6 is over five hours. Similarly, the price will go up as you go up the levels, so in the UK, for example, fees will be from £70 for a DELF2 up to £165 for a DALF 7. The DELF and DALF diplomas are separate certificates so you can choose which one suits your base level best. You can take several tests (i.e. different levels) during the same examination session.
For more information on DELF/DALF and a list of approved assessment centres, you can visit the website of the Centre International d'Etude Pédagogiques.