There is a fish called a fluke, a part of a whale called a fluke and a stroke of luck called a fluke, but these are three separate lexemes with separate etymologies that all happen to share one form. Similarly, a river bank, a savings bank, and a bank of switches share only a spelling.
The first homonyms we ever learn are probably to, too and two (homophones), but the sentence "Too much to do in two days" would confuse no one. there, their, and they're are familar examples as well. lead the metal and lead the verb, or moped the motorized bicycle and moped the past tense of mope are examples of homographs.
In some accents, various sounds have merged in that they are no longer distinctive, and thus words that differ only by those sounds in an accent that maintains the distinction (a minimal pair) are homophonous in the accent with the merger. Some examples are pin and pen in many southern American accents, and merry, marry, and Mary in many western American accents. The pairs do, due and forward, foreword are homophonous in most US accents but not in most British accents. Similarly, affect, effect are distinguished in some careful or cultivated speech.
Homograph disambiguation is critically important in Speech synthesis, but otherwise, homonyms are mostly curiosities, of limited linguistic interest compared to the strong functional roles of antonyms and synonyms. See pun, however. See also polysemy for a closely related idea.
See also: List of English homographs, Initial-stress-derived noun
Homophones commonly confused in the English language
list of commonly confused homonyms