"Slovenly rhyming is one of the sure signs of mediocrity in versification."
—Clement Wood, The Complete Rhyming Dictionary
There seems to be a bit of confusion out there over what exactly constitutes rhyme in English.
But before I can talk about rhyme, I have to talk about syllable stress. [Momentary pause for obligatory stress jokes from the peanut gallery.] Last I checked, nobody was quite sure what exactly stress is, although it's believed to be some combination of volume, pitch, and speed. Nonetheless, native speakers of languages that rely on stress (as opposed to languages that rely on pitch) know what stress is when they hear it. A syllable can be given various levels of stress; the most stressed syllable in a word is said to have primary stress, the next-most secondary stress. The least-stressed syllables are said to be unstressed. (Patterns of stressed and unstressed syllables constitute poetic meter, but that's a topic for another column.)
Look at the word "hieroglyph." It has two stressed syllables, the initial and the ultimate. The first syllable has primary stress (if you're uncertain of this, try saying the word three times, stressing a different syllable each time: HIEroglyph, hieROglyph, hieroGLYPH. Clearly the first of these is closest to the way you'd normally pronounce the word). The last syllable has secondary stress—it's relatively rare in English that two or more totally unstressed syllables appear next to each other (though for the purposes of metrical patterns in poetry non-primary stresses are often ignored, so "hieroglyph" could be considered a dactyl, DA-da-da, even though its last syllable does receive some stress).
Now here's how rhyme works: two words rhyme if all the sounds from a stressed vowel (usually the final stressed vowel) to the end of the word are the same. Notice that I didn't say primary-stress vowel; a secondary stress works too. So "hieroglyph" rhymes with "cliff":
I walked along the rocky cliff
And there I spied a hieroglyph.
Even though rhyming secondary stresses is acceptable, it's usually more satisfying (and often more challenging) to rhyme a word from its primary stress on. For instance, "syllable" more or less rhymes with "full" (to the extent that it doesn't, it's because of the vowel sound, not the lack of stress):
I talked although my mouth was full
Distorting every syllable.
But it also rhymes with "billable":
The doctor's words are billable;
account for every syllable.
Clement Wood's The Complete Rhyming Dictionary is pretty much the definitive source for rhymes and information on rhyme and versification, although the current cheap paperback edition is unfortunately rife with typos. The first section of the book, "The Poet's Craft Book," is chock-full of useful, informative, and strongly opinionated discussion of rhythm, rhyme, and poetic technique; the rest of the book contains over 60,000 entries organized phonologically by rhyming sounds. Unfortunately this organization is mired in an inadequate notation (if only Wood had known about the IPA!), and may seem a bit quirky if your pronunciation doesn't quite match Wood's (he claims, for instance, that "north" and "forth" do not rhyme; in my pronunciation these sounds are either identical or too similar for me to tell apart). Nonetheless, the book is an immensely useful resource for non-free-verse poets and anyone else interested in rhyme.
Wood says that identity is not rhyme. Specifically, he says that two words don't rhyme if they share all the sounds from a consonant before a stressed vowel to the end of the word: "bay" doesn't rhyme with "obey," nor "bare" with "forbear." But Wood may be a bit over-strict in his definitions.
I do agree with him that so-called "sight rhyme"—similarity of spelling but not of sound, as "said" and "paid"—is not true rhyme. Rhyme has to do with sound, not spelling; sight rhyme may be a valid and, in some circles, even respected technique, but it's not rhyme.
On a related note, Wood clarifies something I'd often wondered about: it turns out that words which share all the sounds starting at the consonant immediately after a stressed vowel are said to have consonance. "Willow" and "fallow," for instance, do not rhyme, but are consonant. Emily Dickinson uses consonance quite a lot; it never occurred to me before that this was a specific and intentional device, that it wasn't just a near-miss attempt at rhyme.
Finally, Wood provides a list of final-stress words which appear to have no rhymes in English, such as "fugue" and "pork" and that old favorite "month." Some people regard such lists as challenges. Often finding a rhyme for such "rhymeless" words merely requires distorting a word a little—Christina Rossetti, for instance, wrote
How many weeks in a month?
Four, as the swift moon runn'th.
And sometimes a difficult rhyme can best be accomplished by combining words:
I never knew just who was crosser, us
Or his rhinoceros.
This sort of approach to rhyming has been turned into an art form by such versifiers as William S. Gilbert and Stephen Sondheim. But as Wood points out, such ingenuity is better suited for light verse than for serious.