The Meaning of Communion and the Death of Jesus

Most people who are familiar with church or Christianity have some idea of the relationship between the death of Jesus and the Lord's Supper. But in 1 Corinthians 11, where Paul lays out a proper understanding of communion, what exactly does he mean by the phrase "the Lord's death?," and what does it mean for God's people to "proclaim" Jesus's death when they eat the bread and drink the cup?