In the process of the last supper, Jesus told his disciples This do in remembrance of me. This phrase was understood by the early christians as a command to recreate the Last Supper in a remembrance service.
Initially, the remembrance service took the form of agape feasts. Agape is one of the Greek words for love. This service apparently was a full meal, with each participant bringing their own food, with the meal eaten in a common room. Some echoes of this meal remain in fellowship or pot-luck dinners held at some churches.
This service is known as the Eucharist in Catholic traditions. The name Eucharist is from the Greek word eucharios which means "thanksgiving" or "thank you".
Within many Protestant traditions, the name Communion is used. This name emphasizes the nature of the service as a 'joining in common' between God and humans, due to the sacrifice of Christ Jesus.
A final variation of the name of the service is The Lord's Supper. This name tends to be used by the churchs of minimalist traditions, such as those strongly influenced by Zwingli.