For some of the classic answers to this question, see Why four cups of wine by the seder? and What is the significance of the four cups of wine?

Allow me, however, to share an interesting idea I just came across in the Marcus Lehmann Haggadah. (Rabbi Dr. Marcus Lehman [1831-1890], rabbi and novelist, served as the rabbi of Meintz, Germany, and penned dozens of novels based on historical Jewish figures as well as contemporary stories. Click here to read one of them.)

The numerical value of the Hebrew word for cup, כוס, is 86.

According to out tradition, the worst and most backbreaking period of the Egyptian slavery lasted for 86 years—the last 86 years they spent in Egypt.

Now, the enslavement was supposed to last 430 years. As it is stated in Exodus (12:41): "It came to pass at the end of four hundred and thirty years, and it came to pass in that very day, that all the legions of G‑d went out of the land of Egypt." G‑d, however, in His mercy, began counting the 430 years from the time when He informed Abraham about the upcoming slavery, leaving us just 86 years of actual backbreaking work.1

Thus our slavery was shortened by a total of 344 years.

86 x 4 = 344.

For this reason, we drink four cups of wine, for the four times 86 years which we were spared from slavery.