The word translated 'supper' in verse 20 comes from the Greek word deipnon (Strong's Concordance #G1173). It means the chief or BIGGEST meal of the entire day (usually eaten in the evening) or a feast. Deipnon is translated as 'feast,' meaning a large meal, in Mark 6:21, 12:39, Matthew 23:6, Luke 14:12-24, 20:46, John 12:2 and other places.
At the time of Jesus most people ate only two meals a day: breakfast and dinner. The average Israelite ate foods such as bread, olives, oil, buttermilk, cheese, fruit, vegetables and on very rare occasions meat. Its easy to see that a big, hearty meal was needed after a full day of work in the fields, grinding grain, etc. - especially when the last time anything substantial was eaten was eight or more hours ago. This means Paul could NOT have been referring to a small piece of bread and wine (verses 23-26) as a SUPPER - whether it was the Lord's or anyone else!
So what was Paul saying? He was stating the purpose of the church coming together (verse 20) is NOT to eat a large meal, like the ones they would have at home, or even the meal Jesus ate with his disciples just before his arrest. Such a meal symbolizes nothing and was NOT singled out by Jesus for any special meaning or significance. The purpose of the special meeting (verse 20) is to partake of the simple symbols Christ himself designated and emphasized - unleavened bread and a little wine (verses 23-26) - as a means of commemorating and proclaiming his death.
Was it Communion?
Many in the Protestant world use the term 'communion.' It's one and only connection related to commemorating Christ's sacrifice, found in a few Bible translations, is in 1 Corinthians 10.
10:16 The cup of blessing which we bless, is it not the communion of the blood of Christ? (KJV)
The Greek word translated as "communion" is koinonia (Strong's Concordance #G2842), which means "fellowship" or "social intercourse." This verse is simply describing what the wine and bread represent - a connection or relationship with fellow believers and with Christ - and is not meant to be some title or designation of the two symbols. A more accurate translation of verse 16 of 1Corinthians 10 reads,
16. The cup of blessing that we bless, is it not the fellowship of the blood of Christ? The bread that we break, is it not the fellowship of the body of Christ? (HBFV)
There exists no basis in the Bible for using the terms "Lord's Supper" and "Communion" when referring to what occurred during Jesus' last night on planet earth. When believers eat unleavened bread and drink a small amount of wine each year in commemorating Jesus' sacrifice, they partake of HIM. The correct and truthful phrase for this solemn occasion is 'the Passover' or more properly 'the Christian Passover.'