This is a really carefully organized poem. Let's take a closer look at the first six lines (the first stanza), since what we see happening there gets repeated throughout the poem. Here it is, to jog our memory:
Line 1: Once upon a midnight dreary, while I pondered, weak and weary, A
Line 2: Over many a quaint and curious volume of forgotten lore – B
Line 3: While I nodded, nearly napping, suddenly there came a tapping, C
Line 4: As of some one gently rapping, rapping at my chamber door – B
Line 5:"'Tis some visitor," I muttered, "tapping at my chamber door – B
Line 6: Only this and nothing more." B
Let's start with the rhyming words (we put them in bold to make them easier to see). The first and third lines have a rhyming word at the middle and at the end of the line (as in "dreary" and "weary"). This is called internal rhyme. In the fourth line, the rhyme from the third line shows up in the middle of the sentence (see that "rapping" up there?)
The most noticeable rhyme in the poem comes at the end of the second, fourth, fifth, and sixth lines in each stanza. It's easy to pick out, because it's always an "or" sound (e.g. lore, door, more, floor, Lenore, and of course, Nevermore). That means that two thirds of the lines in this poem end with the same sound! In English-professor jargon, this rhyme scheme would be called ABCBBB, with each letter standing for the sound that ends a line. (We put that up there too to make it easier to see.)
As far as the meter goes, we can start by counting out the number of syllables. The first and third lines have sixteen syllables each. That makes eight pairs of syllables. The emphasis in these pairs is usually placed on the first syllable: (Once u/pon a /midnight/ dreary). This kind of syllable pair is called a trochee. Since there are eight syllable pairs in a line, we call it "octameter" (octo- standing for eight, as in octopus). So the meter is called trochaic octameter.
One thing about having a regular meter, though, is that poets can't resist playing with it. You probably noticed that the last line of each stanza is a lot shorter, with only seven syllables. Plus, if you look really closely, you can see that the second, fourth, and fifth lines only have fifteen syllables. The trick is that in each of the lines ending in an "or" sound, Poe leaves off a syllable. That way the crucial "or" sits out there by itself, unattached to another syllable, making it stand out even more.
If you look, you can probably figure out a few of Poe's other little gimmicks. The point of all this is that he's trying to make his poem as musical, hypnotic, and captivating as possible. All of this complicated rhyme and rhythm aims at drawing you more completely into the world of the poem.