These three terms are extremely important. In fact, they all have to do with whether or not someone who has been trained will be allowed to perform the actions they have been trained to do!
Let's start off by looking at accreditation. If a school has accreditation, they have been recognized by a governing body that they teach a subject well enough that their graduates can be entrusted to perform the jobs that they have been trained to do. This aspect is extremely important because without accreditation, the degrees a school gives out will not be recognized as valid. For example, the Liaison Council for Medical Education (LCME) accredits medical schools in the U.S. If you graduate from a LCME-accredited school, your medical degree is recognized anywhere in the country. If you graduate from a foreign medical school that is not accredited, you need to jump through a bunch of hoops in order to be considered a doctor!
Next, the difference between licensure and certification is a little harder to explain. Both are generally the result of finishing a course of study and/or passing an exam or inspection. However, they have slightly different requirements and purposes.
Certification simply recognizes that a person has been trained and is capable of using that training to do something. Licensure, on the other hand, is the government-given ability of that person to legally perform a certain task. For example, a physician can receive a degree that effectively certifies that he is a doctor. However, he cannot practice medicine until he receives a medical license from the state medical board.
I hope that clears things up!