The Torah uses four expressions to describe our redemption from Egypt: God said to the Jews in Egypt (Exodus 6:6-8):
"I will take you out from under Egypt's burdens - Vehotzeiti"
"And I will save you from their servitude - Vehitzalti"
"And I will redeem you - Vega'alti"
"And I will take you as My nation - Velakachti"
We didn't go from a slave nation to being the Chosen People at Mount Sinai overnight. There were different stages of redemption. The above phrases described these different stages. Each cup of wine represents one of these levels.
There's actually a fifth expression of redemption, "Veheveiti" (and I will bring you into the Land of Israel). This is seen as a reference to the future redemption, to be announced by Elijah the Prophet, when God will gather the Jews from the "four corners of the earth" and return them to their Land. This level of redemption is represented by the fifth cup, called "Elijah's cup," which we pour but we do not drink.