I love this question! I obtained my doctoral degree there back in 2001 and remember my time there with great fondness.
Yale University was founded in 1701 and is the third oldest university in the Americas. I was fortunate enough to be there for the tercentennial. The university has been at its present campus since the mid-18th century. Any institution with that much history has numerous traditions. The biggest sports tradition at Yale is the annual Yale-Harvard football game. The Ivy League created modern American football in the late 19th century, with Yale coach Walter Camp inventing many of the conventions of the game. The Yale-Harvard football game is the oldest collegiate rivalry in the US dating back to 1875. It is always the final game of the season for both teams and even though Ivy League football is generally overlooked now on the national scene, I can assure you that this game has huge significance to both universities. It is referred to simply as "The Game."
A cappella singing groups (no music, only harmonized voices) are also a major tradition at Yale. The gold standard for Yale a cappella groups is the Whiffenpoofs. This group started in 1909 and consists of 14 male seniors. They make regular public appearances and often tour internationally in the summer. Their most famous alum is Cole Porter, a major American musical songwriter in the 20's and 30's.
A more recent tradition is called "bladderball." This is where a large inflatable ball is carried all over campus by different groups trying to gain possession and score points. The rules are rather vague and commonplace property damage caused the game to be officially banned by the university in the 80's (although unofficial revivals have been known to happen). Bladderball was replaced by Spring Fling, a town green music festival near the end of the school year.
Finally, there is a long standing tradition at Yale of not having a commencement speaker (exceptions to this rule are only made for US presidents). The day before graduation does have an event called Class Day, however, that usually does have a big name guest speaker. Hillary Clinton was the Class Day speaker back in 2001 and John Kerry is the speaker this year.
On a personal note, Yale does have a tradition of handing out honorary degrees at every commencement ceremony to notable people in their fields. One of the recipients at my graduation was Steven Spielberg. So now I get to tell everyone that we graduated together!