I think that it is a tabernacle (or so this object is commonly known in the Western Church), for reserving the Holy Mysteries or elements of the Eucharist.
At one time I heard that, generally [or at one point in history?], the consecrated Bread and Wine of the Holy Mysteries were usually only kept in the Eastern Church when there will be a Liturgy of the Presanctified Gifts - i.e., during Great Lent.
I believe that many Churches have the Mysteries so reserved on the altar, in case the need should arise for communion of the sick - something that probably goes back to quite early in Christian history.
Perhaps some others could shed insight on this...
P.S. - Is this what the object looks like? -http://www.easternchristian.com/page189.html#306