Indeed, the story of humanity begins with marriage.
When God creates Adam, God states: "It is not good that Adam is alone--I shall make a match for Adam" (Gen. Despite the importance of marriage, the Bible makes no mention of the wedding ceremony.
The ceremony that accomplishes nisuin is known as chuppah.
Today, erusin/kiddushin occurs when the groom gives the bride a ring or other object of value with the intent of creating a marriage.
At a traditional Jewish wedding, the bride's side is on the right (if you're facing the chuppah) and the groom's is on the left.