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.
One application through uni-assist costs 75 EUR and each further application costs 30 EUR. In the past refugees were able to apply to 3 universities for free but this special regulation is ending on the 31.12.2019. Inform yourself about the fees here.