Today is February 29, the leap day of the current leap year. Reason enough to have a quick look at the terminology employed in various languages.Usually the designation for the day is by metonymy extended to the year in which such a day occurs. For instance, if the leap day is called “extraday”, a year with such a day is usually called “extrayear”, even though it is not an entire year that is extra.

Leap day

This refers to the fact that the leap day leads to certain important days “jumping one day ahead” in leap years. Let’s take Christmas Day, for instance. In 2010, it was on a Saturday. In 2011, on a Sunday. But in 2012, it will be on a Tuesday, it “leaped over” the Monday. In 2013, it will be a Wednesday.

  • English: leap year and leap day.  (Also, bissextile year and intercalary year)
  • Dutch: schrikkeljaar and schrikkeldag. schrikken, even though it also means “to be scared”, it can also mean “to take a leap” (it seems though that nowadays this meaning seems to be obsolete). (Source)
  • Malay uses a calque from English: tahun lompat, hari lompat. But also see Indonesian in the next section.
  • Polish seems to be using a word that means “step over” (Rok przestępny), but my Polish is very rusty. Just wanted to at least have one Slavic language on my list. 

Inserted day (“Intercalary day”)

The idea is that the day is inserted into the calendar (Latin intercalare), which is why another name in English for it would be intercalary day

  • German: Schaltjahr und Schalttag, from schalten. Why this word nowadays means “switch” and could be understood in the sense of “switching on” a leap day, this explanation would be anachronistic as there were no electric appliances in the Middle Ages. It turns out that this word also meant “push” which it no longer does in Modern German. So the day was one that was “pushed in”, i.e. inserted.
  • Swedish, Danish, Norwegian: the Scandinavian languages all use forms of the verb “shoot”: skottår/skottdag, skudår/skuddag and skuddår/skuddag. In the modern languages, the semantic link is broken again. In Sweden, people sometimes wonder if it has anything to do with Scotland, which would be Skottland in Swedish. But again, the answer is that is a loan translation from the German Schaltjahr, and for instance in Swedish, skjuta can not only mean “shoot”, but also “push”, and we have the same “push in” pattern here. (Source)
  • Indonesian: tahun kabisat/hari kabisat. "Kabisat" comes from Arabic and means something like "pressed/stuffed", and this is the term generally used in Arabic (and other Islamic cultures like the Persian one) for "intercalary".

Additional/residual day

In some languages, the leap day just means something like extra day. I think this is also the case in Esperanto, where the leap day is supertago and the leap year superjaro. So the designation carries over to the year in the sense of “year that has an additional day”.

  • Turkish has Artık yıl and Artık gün, where artık means “rest, residue”
  • Hindi has अधिवर्ष adhivarṣ(a) and अधिदिन adhidin(a) directly from the Sanskrit, which I presume is a contraction of अधिक वर्ष adhika varṣa "additional year" and  अधिक दिन adhika dina "additional year" respectively   (adhi- can also be a prefix, but the intercalary month is अधिक मास adhika maasa after all).

From ancient Roman tradition

In Ancient Rome, the year was called annus bisextilis in Latin, referring to the day that was repeated, the sixth day before the calends of March, i.e. February 24th (the Romans had a very peculiar way of counting their days in the calendar), so it was ante diem bis sextum Kalendas Martii. Even though the leap day no longer falls on February 24th, many Romance language still use this term.

  • French, Spanish, Italianannée bissextile, año bisiesto, anno bisestile.

From ancient Chinese tradition

Chinese, Japanese and Korean use words that go back to a very ancient Chinese tradition almost no-one is aware of synchronically. Back in the day, of course, the Sinosphere was still using the traditional lunar calendar, which also had intercalary months and days. It was customary in ancient times that the ruler of a state would not go about his normal duties during such an intercalary day and rather stay inside the gates. From this, a character was formed, 王 “king” + 門 “gate” = 閏 rùn. 

  • Chinese 閏年/闰年 rùnnián 閏日/闰日 rùnrì.
  • Korean 윤년 yunnyeon and 윤일 yun’il 
  • Japanese  accordingly also has 閏年 and  閏日. They can be read in Sino-Japanese as junnen and junjitsu, but there is also a kun’yomi reading as uruudoshi and uruubi, which is actually more common. This is also due to confusing the obscure 閏 character with the more common 潤 character meaning “moist”, and which has uruou as one of its Japanese readings.

(Sources in Japanese and Chinese)