The name 'Shavuot' means 'weeks' in reference to the completion of the seven week counting period between Passover and Shavuot. Shavuot is primarily the anniversary of the giving of the Torah (Teachings) to the Jews.
Additionally, Shavuot is when we celebrate the first fruits of the season and the wheat harvest. As such, Shavuot has two other names in the Tanach (Jewish Bible): Yom Habikurim (Day of the First Fruits) and Chag HaKatzir (Harvest Holiday).
Please see the related articles for more information.
Every one of our festivals has more than one name and purpose.
Each of them has as its purpose "remembering the Exodus from Egypt"
(as stated in our prayers and the kiddush over wine). In addition,
Passover is a thanksgiving to God for the barley-harvest, Shavuot
is a thanksgiving to God for the wheat-harvest, and Sukkot is a
thanksgiving to God for the ingathering of grain.
Shavuot also celebrates the Giving of the Torah on Mount Sinai, and Sukkot commemorates God having protected us in the wilderness.
It may also be noted that it is instinctive and a moral and emotional need to celebrate in front of God every so often. This was Cain's motivation in making his offering in Genesis ch.4 without having been commanded.
Had God not given us the Torah-festivals listed above, we might instinctively seek out those of the Canaanites, which the Torah warns against (Exodus 34:15) immediately before listing the Jewish festivals (in the following verses).