Introduction
Thinking about the Lord's Supper is complicated by the largeness of the theology involved. What you think about the person of Christ intersects with which parts of soteriology you focus on, both of which are colored by how much weight you give to eschatology and all of which is overlaid on how you believe the Old Testament is fulfilled, let alone questions about actual church practice!
Tangibly we eat and drink in context; within a tradition, Passover, after hearing some sort of explanation and communicating our thanksgiving through prayer. This makes sense of the remembrance bit, otherwise it'd be eating and drinking in a vacuum, reading whatever emotional experience we wanted into the Supper. So using only the texts that speak primarily on this topic but drawing on larger systematic themes, I think the Lord's Supper does these three things intangibly.
Promise:
We can't see the future but it's real. The Lord's Supper even in it's highly ritualized form is the beginning of this final feast. That's why when we eat we proclaim the Lord's death until he comes. (1 Cor 11:25) This also makes sense of Jesus remarks at the Last Supper; "from now on I will not drink of the fruit of the vine until the kingdom of God comes.” (Luke 22:18) So it's as though we nibble on a little of the future.
Assurance:
Jesus describes the wine of Communion as symbolic of my blood of the covenant, which is poured out for many for the forgiveness of sins. (Matthew 26:28) The atonement, a vicarious sacrifice is a singular event with spiritual consequences, true justice and eternal hope. Communion doesn't recreate any of these effects but comforts and reminds us of our connection to the “new covenant" (1 Cor 11:25, Luke 22:20).
Unity:
If we are Christians then we are united to Christ, this is why when we partake of the table of the Lord (1 Cor 10:21) we partake of Christ. This then also explains the warnings about properly discerning the body of the Lord, he isn't a plaything. (body of the Lord ... judgement 1 Cor 11:27-34) This union however isn't visible to the naked eye so it's expressed invisibly when we remember together.
[Random iStock photo of a table inside a church.]