Pronunciation of the letter “e” in French changes when it belongs to a larger group of letters or when it is supposed to be silent—you can read more about that in my previous article. It can also be modified by various diacritical marks. If this is not the case, we can (usually) determine the pronunciation according to the following rules:
E in open syllables
In monosyllabic words, such as je, ne, etc., the pronunciation depends on how much you want to articulate; it is common to keep the “e” silent and pronounce, for example, je ne as /ʒn/, but you can also try to sound more articulate by pronouncing it /ʒənə/. This is probably the pronunciation they have taught you in school (if you learned French there), but the first one is more common in everyday speech.
This takes us to the second possible pronunciation of “e”, namely /ə/ (as “a” in the name Tina). It is pronounced this way in open syllables (i.e. syllables that don’t end with a consonant); for example, gouvernement (government) is pronounced /guvɛʀnəmɑ̃/. The word consists of four syllables: gou-ver-ne-ment. The third syllable is open, i.e. “ne” is pronounced /nə/, while the second syllable, “ver”, is closed (i.e. it ends with a consonant), and so it is pronounced /vɛʀ/ (i.e. the “e” sounds as in “bet”), which we will explain in the next paragraph. Similarly ressembler (to ressemble) is pronounced /ʀəsɑ̃ble/, because “re” is an open syllable. However, as with words like je, ne, etc., the /ə/ sound can usually be dropped in everyday speech, e.g. gouvernement and ressembler can be pronounced also as /guvɛʀnmɑ̃/ and /ʀsɑ̃ble/, respectively.
E in closed syllables
The last case we have to deal with is closed syllables. We can say that “e” in a closed syllable is pronounced /ɛ/ (as “e” in “bet”), unless ‘dictated otherwise’ by groups of letters it appears in. For example, maternel (maternal) is pronounced /matɛʀnɛl/ because both “ter” and “nel” are closed syllables. There are some obvious groups that change the pronunciation, such as “em” and “en” being pronounced with a nasal “a”, as in mental /mɑ̃tal/. However, there are also some other, more subtle changes. There are several cases in which “e” is pronounced /e/ (as “e” in “hey” in English).
This happens mostly with the “-er” ending of an infinitive, “-ez” ending of a word, or a monosyllabic word that ends in “es”. For example, parler (to speak), laver (to wash), etc. are all pronounced with /e/ at the end (“-r” remains silent in infinitives). “-ez” can appear as a part of ordinary words, such as nez /ne/ (nose), or as the ending of the second person plural of verbs (such as parlez /paʀle/ ([you] speak) (i.e. “parler” and “parlez” are pronounced exactly the same).
It is also important to pronounce words like les /le/ (the, plural), ces /se/ (these, masculine), etc. correctly to distinguish them from the words le /lə/ (the, masculine singular), ce /sə/ (this, masculine), etc. Note: c’est, meaning “this is”, should be pronounced /sɛ/ in standard French, but in everyday speech it is often indistinguishable from ces. Similarly, es ([you singular] are) should be pronounced /ɛ/, but the pronunciation /e/ is common.