Hippopotamus is a Latin word that has been imported into English. In Latin, its plural is hippopotami. In English, we sometimes still use the Latin plural. Like many unusual and interesting English words, it was first coined in Greek and later absorbed into Latin. In Greek, it was hippopótamos, with plural hippopótamoi.
How to think about this
Latin and Greek plurals are primarily jargon used in scientific or academic work. For example, antennae, the Latin plural of antenna, is preferred in entomology, but the English plural antennas is more common in popular and informal writing.
However, many Latin and Greek plurals also have popular usage: basis/bases, memorandum/memoranda, and many others. These often carry a feeling that they are not fully native words: we're "borrowing" them from Latin and Greek. The more the word seems like a fully adopted member of the English language, the more likely we are to give it an English plural. For example, as a plural of index, the English-style indexes has a long-running competition with the Latin indices. In some cases, like basis, very few people use a native plural, maybe because it would sound too weird (basises—hmm, sounds a little like hippopotamuses).
To fully understand this, you have to know the basics of the history.
From about 800 to 1800 C.E., Latin was the international language of Europe and the main language of science, law, administration, scholarship, and religion. Latin had several advantages that made it well suited for these roles. It was nobody's native language, so no one country "owned" it. If you went to a university, you learned Latin, and you spoke in Latin at the university, at least during classes and lectures—even in your native country. Latin was defined by a written tradition going all the way back to ancient Rome. This made Latin remarkably stable. While its vocabulary grew, its core changed very little over the centuries—much less than vernacular languages that people learn as their mother tongue.
Latin words came into English in two ways. One way was through the conquest of England in 1066 by the French-speaking Normans. French grew out of Latin, and English absorbed a huge number of words from it. The other way was by borrowing directly from Latin when you needed precise, specialized jargon not subject to the haziness and fluctuations of everyday language. If Latin didn't have a word, you borrowed it from Greek. Today, the majority of English words come from Latin (or Greek, usually via Latin).
The period of the rapid expansion of English vocabulary is long past, as is the tradition of most educated people learning Latin and Greek. However, we still exploit Latin's stability and internationality in the scientific names for species and in a lot of scientific, academic, and legal jargon. And we still retain the tradition of indicating word origins in spelling—and a little bit in grammar. (We even have a few Hebrew and Italian plurals in English.)
Latin plurals are more complicated than native English plurals. Latin nouns fall into several different patterns, called declensions. Hippopotamus follows the "second declension", like stimulus/stimuli, nucleus/nuclei, fungus/fungi, and others. But corpus and genus are "third declension", with plurals corpora and genera. And apparatus is "fourth declension", with plural apparatus. Some "first declension" Latin nouns in English are larva/larvae and differentia/differentiae. The other main patterns are quantum/quanta, curriculum/curricula, bacterium/bacteria, and appendix/appendices, vertex/vertices, matrix/matrices.
I've been searching for the past 15 minutes and I still haven't found a good source of information about these for an English learner, but this might be helpful. You don't need to know which declension is which. You get familiar with each plural after you run into it a few times, and that's good enough.
The -ii plural ending only comes up when the singular ends in -ius, as in radius/radii and denarius/denarii (which are the only two I can find).
The plural of virus is viruses and the plural of bonus is bonuses. These words do not have Latin plurals in English (although people sometimes enjoy pretending that they do). If you'd like to know why they don't, that's probably a good question for ELU. A short answer that skips many details is: viri is Latin for "men", not "viruses"; and boni in Latin means "good men", not "bonuses".
How to choose
As shown on this graph, hippopotami was much more common than hippopotamuses for most of the last 300 years, but today they're about equally common. Some people would say that this proves scientifically that both are now equally "correct" or "acceptable". I hope that you won't think that way. I consider that pseudo-science. When using a Latin plural, you should simply be aware that that's what you're doing. As you find Latin plurals used, or not used, in different situations, you should develop some feeling for how different people in different situations perceive them.
You should get a sense that the word hippopotami is fun to say and sounds a little pedantic, but hippopotamuses is also fun to say and can sound a little silly since a six-syllable word ending in -us seems to cry for a Latin plural and the unstressed -amuses is unusual (amusing, even). You should choose which form to use depending on the effect you want to produce in other people, the situation, your mood at the moment, your audience, and any other reason you like—exactly as you should choose any other word. But you're a zebra, so you already know all that.