Homonym, homophone, homograph
If you look at noders' writeups for these three words, and the
Webster 1913 definitions, and then some other
dictionary that you trust more (in my case, any that I wouldn't burn on the spot), you'll find we have considerable confusion on these terms. At a recent gathering I was asked to settle the matter, and had to admit I didn't know, I always had to look them up. When I got home I did look them up in a dictionary of
linguistics and sort of went
hmph. The noise you make when you've got half an explanation and it's something like what you thought but the other half is not going to come any time soon.
These words just are used in overlapping senses and you mustn't try to force a rigid distinction between them. But here's how their etymology can suggest a good way of using them as three distinct terms. The Greek roots are of course hom(o)- 'same', onym- 'name', phon- 'sound', and graph- 'writing'.
In this suggestion, a word is a homonym of another if the two things have the same name. The same linguistic token is used for both. This is the more general term, because the linguistic token can be either the way it's spoken or the way it's written. A word is a homophone of another if they have the same spoken form. A word is a homograph if it has the same written form as another.
Under these definitions we see that a homonym can be either a homophone or a homograph or both. Examples of homophones (same sound) are: bear the animal and bear to carry; or beer and bier. Examples of homographs (same writing) are bear the animal and bear to carry; or lead the metal and lead to direct. All those could be called homonyms. Where both spelling and pronunciation coincide you could call them homonyms in a narrower sense; and if they are etymologically related it is called polysemy.
But it is also natural to use narrower definitions. Two words are homophones if they have the same sound and by implication different spellings: such as beer, bier or passed, past. In the same way two words are homographs if they are written the same and by implication sound different, such as lead (metal; direct) or bow (weapon; greeting).
As far as I can tell, the words are commonly used in both the wider and the narrower acceptations, and I am not aware of a strong preference in linguistics for either. And homonym is used as a variably general term, so if in doubt use that and explain what you mean.
Homophony is of course dependent on dialect. In half the dialects of English caught and court are homophones, and in some of the others caught and cot are. Some dialects keep a short unstressed E or I, and others turn them into the same sound as unstressed A: so imminent, immanent or allude, elude will be homophones or not depending on how you speak. Some accents distinguish prince, prints. An isolated example is boy/buoy, which are not homophones in American. Regional variation in homographs is harder to find but one example is metre the unit of length and meter the rhythm of a poem, both written meter in the US.
On polysemy, it is also partly a question of interpretation whether you distinguish two words or just say the same word is being used in two ways. We get into etymology. Fair the market is Latin and fair the hair colour is Germanic, so they're definitely different. But fair maid and fair distribution both come from the hair colour, via (roughly) this kind of sequence: blond -> clean -> beautiful -> even, balanced, proportionate -> just, equitable. Even knowing this etymology, these still feel like all different words. But what about rough? The senses of 'not smooth' and 'approximate' (and of course 'the worse for wear') seem close and we might want to call them variant usages rather than homonyms. Related to this are conversions, such as 'to waitress', verb, from the noun 'waitress'. The distinction between polysemy and coincidental homophony is important in lexicography because dictionaries sometimes group polysemic words (polysemes?) together under the same head.
For a second opinion see the Columbia Guide at www.bartleby.com/68/21/3021.html because they more or less agree with me, qualifying their examples with "and usually" and "and some would add", and talking about "one sort" of homonym and "another sort".