Eucharist: From the Greek Eucharistia, meaning thanksgiving. Eucharist was at first the name for the prayer of thanksgiving that preceded the transformation of the bread and wine into Christ's Body and Blood in the liturgy of the early Church. Later the term was applied to the whole celebration of the Mass.
Communion: From the Latin communia, meaning fellowship. By Communion, we mean recieving the Body and Blood of Christ in the transformed (consecrated) gifts of bread and wine. This occurs as a rule during Holy Mass, but also on certain occasions outside of Mass (when brought to the sick). Communion only under the species of bread is also a full communion with Christ.
Source: Youcat sidebars (Pgs 123,127)