Erasmus is a student Exchange Program among European universities. You can study at another university for one or two semester. The credits are usually transferable. It is now open to foreign students, too, in Semmelweis University, although it has just began for foreign students.
It depnds on the host University and your preference, but it seems a host Univeristy usually accepts only for one semester, in case of Semmelweis univeristy. There is a bilateral agreement between host and home university.
Many universities are involved. It depends on your home university;i.e. if nobody want to come from your desired univeristy to your home university, you can not choose that host university, because that is how a bilateral agreement is formed, namely "give and take".
For Example, there was Karolinska Institute(Sweden) on the list. But it was deleted because nobody from the institute would like to come to Budapest.
Some examples from the list.