What is the climax of "The Raven"?

The climax in this poem serves to illustrate how the speaker dwells on the loss of his loved one, Lenore. And some would say he dwells on this to the point of torturing himself. This is a narrative poem, so we can treat it like a story with elements such as exposition, plot, and climax. 

In the exposition, the speaker is reading from "a quaint and curious volume of forgotten lore" in order to take his mind off of his grief. He begins to hear a tapping on his chamber door. He investigates it and finds nothing. This "nothing" symbolizes the absence of Lenore. Then the raven flies into his chamber and perches on the bust of Pallas/Athena. The raven represents death and darkness. Positioning itself on top of a symbol of reason and logic, the raven imposes these notions of death and darkness onto logic. The speaker dwells on his sorrow and the death of Lenore; his reason is suppressed. 

The plot rises as he continues to question the raven. He is looking for answers, but he knows what the response will be. He wonders if he'll ever be able to forget Lenore. He says "Quaff, of quaff this kind nepenthe, and forget this lost Lenore!" To quaff is to drink and nepenthe induces forgetfulness. He wants to know if he'll ever be cured of his sorrow. He says "Is there--is there a balm in Gilead?" This is a balm that will cure illness. In this case, his illness is his grief. The raven responds "nevermore" to both questions. The climax occurs in the third to last stanza. He directly asks if he will ever see Lenore again, even in heaven (Aidenn). The raven again responds, "Nevermore." This is the climax and it is heightened by the fact that the raven will not leave. Thus, the speaker is faced with never seeing Lenore again and with the raven staying as a reminder, he will never forget his sorrow. 

When is the conflict resolved in "The Raven"?

The conflict that exists within the speaker who has lost his love, Lenore, is resolved when he finally despairs of ever being reunited with her.

The forlorn lover utters these lines at the poem's end:

And my soul from out that shadow that lies floating on the floor
Shall be lifted—nevermore!

In addition to its marvelous and innovative verse, Edgar Allan Poe's poem "The Raven," evinces a macabre tone that crescendos with these last lines. For, the verse begins in a relatively light tone as the speaker seeks "surcease of sorrow" in his books; however, after he responds to a light tapping at his door and allows the strange visitor, a raven, to enter, the tone becomes ominous. For instance, the speaker begins to become perturbed and expresses his agitated state of mind with the alliterative use of the words grim, ungainly, ghostly, and gaunt. Further in the poem, the speaker becomes very unsettled as he perceives the dark bird as a symbol of lasting sorrow:

This I sat engaged in guessing, but no syllable expressing
To the fowl whose fiery eyes now burned into my bosom's core:
This and more I sat divining....

Just as the ominous bird refuses to leave, so the terrible sense of grief and loss remains in the heart of the speaker, to be lifted from him "nevermore."