Communion: From the Latin communia, meaning 'fellowship'. 'Communion' or 'The Lord's Supper' are two names for the sacrament where Jesus' followers remember his death on the cross by eating bread and drinking wine. There is a lot of debate among different Christian denominations as to what exactly communion means and how it should be carried out.
Eucharist: From the Greek Eucharistia, meaning 'thanksgiving'. Eucharist originally referred to the prayer of thanksgiving that preceded communion, but now is often used for the whole event.