Heteronym, Homograph, Homonym, and Homophone

A homograph is a word that has the same spelling as another word, but has a different meaning, such as lead (to go in front of) and lead (a metal). The ending –graph means drawn or written, so a homograph has the same spelling.

Heteronyms are a type of homograph that are also spelled the same and have different meanings, but sound different, such as above or bow (tied with ribbon)
bow (of a boat).

A homophone is a word that has the same sound as another word, but is spelled differently and has a different meaning, such as to, two, and too. The ending –phone means sound or voice, so a homophone has the same pronunciation.

A homonym means either a word that is spelled like another, but has a different sound and meaning (homograph) or a word that sounds like another, but has a different spelling and meaning (homophone).
OR
A word that is spelled and pronounced like another, but has a different meaning (homograph and homophone), like by (near) and buy (to purchase).

Strictly speaking both homographs and homophones are homonyms, but homonyms can be either or both a homograph and homophone. Heteronyms are always homographs, but homographs are not always heteronyms.

Discreet vs. Discrete

This pair of homophones (words that sound alike, but are different in meaning, spelling, or both) can be confusing. Discreet implies the showing of reserve in behavior or speech. Discrete means distinct, separate, unrelated.

Both words derive from the same Latin word discretus meaning “separated.” Until the 1700s, these words were each spelled many different ways including discrete, discreet, dyscrete, discreete, etc.

Eventually discrete and discreet came to be differentiated in spelling as well as in meaning. Discreet has yielded the noun discretion, but discrete's noun form is discreteness. For most of English history, discreet was more frequently used, but today discrete is much more frequently used than discreet; it has seen a dramatic rise since the 1940s.

If the e’s are separated by the “t”,  use “discrete” (meaning “separate”).

Discreet vs. Discrete

Discreet describes showing “reserve, prudence, or cautiousness” in one’s behavior or speech. The noun form of discreet is discretion. Both discreet and discrete derive from the Latin “discretus”, meaning separate, situated, put apart, which derives from the Late Latin discernere (where the word “discern” came from).

Discrete means “distinct, separate, or unrelated.” The noun form of discrete is discreteness.

Here is how each might be used in a sentence.
These two items are discrete.
The politician was not discreet.

Discrete and discreet are homophones; words that sound alike, but differ in meaning or spelling or both.