What is an “abortion on request”?

That is the case when legally it is not required to provide any reasons or to explain the circumstances of your decision to have an abortion. However, there still might be other requirements that you have to fulfill in order to access abortion, such as parental or spousal consent, compulsory counseling or mandatory waiting periods. 

Some countries allow an abortion on request only up to a certain length of a pregnancy, and require other grounds (such as the endangerment of health or life of the pregnant person, fetal impairment, rape, etc) for an abortion after that gestational limit. The gestational limits and grounds vary from country to country, and sometimes they are not further specified in the law at all.

For more detailed information you can consult the Global Abortion Policies Database.