The 16th century kabbalist, Rabbi Isaac Luria, known as the Ari — a Hebrew acronym for the “Godly Rabbi Isaac” — established the custom of learning Torah all night on the Shavuot eve. A tikkun is a correction. In kabbalistic circles it refers to a spiritual correction for some fault. There are several types of tikkunim. For example, Tikkun Hatzot (midnight) is a midnight prayer mourning the destruction of the Temple and yearning for its rebuilding.
According to the Midrash, the children of Israel overslept the morning of the giving of the Torah and God had to rouse them with a shofar blast. As a tikkun, the Ari instigated the custom of all night learning. Where there had been too much sleep, there was now no sleep.
Leil is an abbreviated form of leilah (night). Instead of saying haleilah shel Shavuot (the night of Shavuot)one can condense this mouthful into leil Shavuot. Leil has no meaning when not connected to a noun, such as Shavuot or Shabbat. It can be painful for Israelis spending time in England to hear people say, “Are you going to the Tikkun Leil?” The word “Shavuot” in this phrase is not optional.