Key to understanding this famous poem is establishing that it is the speaker of the poem that is creating the problems he is suffering for himself. Poe himself wrote that he was exploring through this poem one aspect of the dark side of human nature - "that species of despair which delights in self-torture." In other words, the narrator projects or puts onto the bird whatever his own wild imagination dredges up.
In terms of the conflict then, the conflict in this poem is decidedly an internal one, as we are presented with a frail and exhausted student working late at night mourning the loss of his love, Lenore. It is clear from the description in the first stanza that he is not in his right state of mind: he describes himself as "weak and weary" and tells us that he has dived into study and books to try and "borrow/From my books surcease of sorrow" for the "lost Lenore". He is basically trying to desperately forget his love for her and get over his grief. Thus, when the Raven appears, the narrator believes that this is some kind of external conflict between himself and the Raven who the narrator views as a messenger from hell or a demon who has come to taunt him with the impossibility of ever getting over his grief. However, what the narrator does not realise is that it is he that is driving this conflict - he is the one who is making the suggestions about the Raven. The Raven only gives one word in response, which the narrator interprets from his own perspective. For example, consider this stanza:
"Prophet!" said I, "thing of evil! - prophet still, if bird or devil! -
By that Heaven that bends above us - by that God we both adore -
Tell this soul with sorrow laden if, within the distant Aidenn,
It shall clasp a sainted maiden whom the angels name Lenore -
Clasp a rare and radiant maiden whom the angels name Lenore"
Quoth the Raven "Nevermore."
The narrator, knowing that the Raven has only spoken one word, asks it a question phrased in such a way that the inevitable response will plunge him ever further into his self-made abyss of despair and grief. It is this that reflects the true conflict in the poem which is only resolved by the plunge of the narrator into his despair from which he feels he can never escape.