Public institutions in South Africa each run their own application process. There is no single national form: you apply to every institution you are interested in, separately, and each one sets its own opening and closing dates.
Before you start
Have these ready as scans or clear photos, because almost every application asks for them: your ID or birth certificate, your most recent school results, and proof of your postal and residential address. If you are applying for funding at the same time, you will usually also need proof of household income.
Apply early, apply widely
Applications for the following academic year usually open months before you write finals, and popular programmes close first. Applying to more than one institution costs a little more in application fees but keeps your options open when results come out.