Uni-assist is an organization which processes the applications of international students to German universities. Uni-assist works with most German higher education institutions. You can find the answers to many questions you might have about them in their FAQ section.
If you want to study at one of the 180 German universities that use uni-assist, you need to apply through uni-assist instead of through the university. You can find a checklist of the application procedure on uni-assist's website (scroll down for English).
International applicants normally need to pay a fee to uni-assist for checking the documents. If you are a refugee, however, you don’t have to pay this fee and can apply to 3 higher education institutions/pro semester for free. To get this cost exemption for refugees, you have to apply on uni-assist with either a TestAs or a Hochschulbescheinigung. More information on how to do so can be found on uni-assist's website (scroll down for English).