Octopi, the supposed plural of octopus, is a favorite among fans of quirky words, but it has no etymological basis. The form was created by English speakers out of a mistaken belief that octopus is Latin and hence pluralized with an -i ending. But octopus comes from ancient Greek, where its plural is octopodes, and though it came to English via scientific Latin—one of the late varieties of Latin that kept the language alive long after it had died out as a first language—it was never a native Latin word and didn’t exist in that language until scientists borrowed it from Greek in the 18th century (and if it were a Latin word, it would take a different form and would not be pluralized with the -i ending).
All of that is beside the point, though, as octopus has been in English for centuries and is now an English word when English speakers use it, so there is no reason not to pluralize it in the English manner. Granted, some Latin and Greek plurals survive in English by convention, but octopi/octopodes is not one of them. Octopuses is far more common than octopi in edited writing of all kinds, including scientific writing.
Still, while the use of octopi can’t be justified on an etymological basis, it is not wrong. It is old enough and common enough to be considered an accepted variant.
People love octopi, so the quirky plural appears often—for example:
The former is served a martini glass, holding about a dozen mini octopi cooked in a sweet and spicy sauce … [Birmingham News]
A ban on fishing in the region may affect the market for octopi and marine-based food products popular in Los Angeles … [LAist]
Noise pollution knocks squid, octopi off balance [MSNBC]
But most edited publications use the boring but perfectly acceptable octopuses—for example:
Octopuses are highly intelligent animals and have been proven to have a strong short and long-term memory. [Telegraph]
Octopuses, turtles and rays glide along the seabed. [Sydney Morning Herald]
Cable had a great year, and media octopuses like Time Warner and News Corporation continue to find plenty of profits. [New York Times]
Octopodes appears occasionally, but it’s liable to cause confusion.