This almost certainly refers to vaccination against smallpox. The term "vaccination" originally referred specifically to inoculation with the cowpox virus ("vacca" being Latin for "cow"). Cowpox is related to smallpox, but causes a much milder disease. Because of its similarity to smallpox, infection with cowpox also confers immunity to smallpox. This was the first vaccine to be developed. From the introduction of vaccination against smallpox in the 1790s until the early 20th century, smallpox was the only disease that was regularly treated preventatively in this way. So, for the whole 19th century, "vaccination," unqualified, referred to smallpox.

It also appears that this vaccination was mandatory. Why have a certificate otherwise? This certainly looks to have been issued by an agency of the government. Vaccination against smallpox was made compulsory in Victoria in 1854; a series of Health Acts followed, including the one in 1890 that is referenced by this document. No other immunizations were required in the 1890s, because no immunizations against other common diseases were in wide use at the time. A vaccine for rabies had been successfully tested (though it was used after exposure, not as a preventative), and work was underway on others, but there was no question of mandating vaccines against anything else until shots for diseases like pertussis and diphtheria were introduced in the 1920s.

There's an additional clue here. The certificate says that your ancestor was "successfully vaccinated." The smallpox vaccine is not injected with a syringe. Instead, the skin is scratched or pricked several times with a pointed needle (today the needle is bifurcated, or forked, though that was not always standard) that has been dipped into a vial containing the vaccine. Over the next few days a pustule forms, which eventually dries into a scab that leaves a small scar. Many people who were born before the 1960s still have this scar, usually on their upper arm. But occasionally a pustule does not form. This indicates that the patient is not responding to the vaccine, likely because the vaccine was either expired or incorrectly administered. If this happens, the patient has not been immunized against smallpox. So, in cases where vaccination is required, the patient's arm needs to be inspected after a few days later to check for a pustule before a certificate like this could be signed.

Edit: I'm on mobile and didn't notice that there is a second image. The language about compulsory vaccination and inspection confirms that this is smallpox.

This doesn't specifically answer your question but here you can read the entirety of Australia's 1890 Health Act (Part IX: Vaccination), there may be some insights into what might have been going on. At that time, smallpox, rabies, and Cholera (among others) were being tackled by most developed nations. Your local bureaucracy apparently didn't see fit to include what was being vaccinated against on the record of vaccination itself, for some reason... It's hard to tell from this document what exactly was going on, but there were many mass-vaccinations going on at the time. Its popularity was growing rapidly. A search of Australia, or that area's, history at that time may tell you which diseases were most prevalent, and might narrow down the possibilities. What, specifically, are you trying to find out?