A dramatic monologue is a type of poem that takes the form of a speech (not the poet, but of an unnamed narrator) which tells a story. The speech is directed at a specific audience, but the reader must determine from the context who the audience is and why the speaker is telling them this. T.W. Eliot's "Journey of the Magi" is certainly a dramatic monologue, as it relates the story of one of the magi (wise men) who traveled to see the baby Jesus. He is old now, and has seen the "first dispensation" (the old law for the Jews) pass, and a new belief system take hold.
And allegory is an extended metaphor used to convey hidden or complex meanings through symbolism. This poem may be seen as an allegory, as well, because it symbolically refers to the birth, life, and death of Jesus and the change he brought to religion. The most obvious references are in the middle stanza, although the last clearly brings Birth and Death into the the picture together, as well. Consider: when the magi come into the valley, they see "three trees on the low sky" (an image of Golgotha), "and an old white horse galloped away" (Rev 6:8), a "tavern with vine leaves" (Jesus's blood is "the fruit of the vine"), and "dicing for pieces of silver" (possible reference to Judas's betrayal for 30 pieces of silver).