We believe that Christ instituted the Eucharist as two species to symbolize the separation of his body and blood as in his death on the cross. For this reason both species must be present at the mass and at least the priest must consume the sacred contents of the chalice.
However, Christ is not half present in one species and half present in the other. He is fully present, Body, Blood, Soul, and Divinity in the host as well as fully present, Body, Blood, Soul, and Divinity in the contents of the chalice. For this reason it is not necessary that every person recieve under both kinds as he will recieve the whole Christ under either species. However, if you feel called to recieve under both kinds, that option is readily available in many parishes.
I know that Byzantine Christians always recieve under both kinds. But, do they believe that Christ is fully present under each species or must both species be recieved in order to recieve the whole Christ?