The only requirement for the student is to be in possession of an identity card when crossing borders within the European Union. This is made possible by the free movement of persons within the EU member states and the EFTA countries.
However, you must register at the municipal Population Department of your Erasmus destination for registration in the Immigration Register. On the contrary, there are no general regulations regarding the documents which have to be submitted for this purpose.
For studying abroad outside Europe, you need in almost every case a (student) visa.
A visa is a document which allows you to travel and to stay in a certain country. The visa is issued by the embassy or the consulate of the country of destination; or at the border. The visa is very often issued in the passport.
A couple of conditions are bonded to receiving a student visa (among others: the submission of a certificate of enrolment to an establishment of higher education in the country of destination). Please check what the conditions are, and make sure that all documents are available at the time of application. For more information: www.diplomatie.belgium.be.