א is a glottal stop, like the break in the phrase "uh-oh". It's a closure or clicking of the glottis, which is the lowest opening in the throat used for speech, right above the voice box.
The correct pronunciation of ע is a voiced pharyngeal fricative. The simplest way to describe the sound to an English speaker, in my opinion, is by squeezing the throat and using your voice at the same time. A good way to practice this is by bunching up your tongue as far back as you can until you feel your throat’s opening get narrower. Try to replicate that feeling in your throat, but keep the tip of your tongue behind your teeth, and breathe out while using your voice.
Sources: ספר היצירה, דקדוקי הטעמים, short Hidāyat al-Qāriʾ II.§1.¶2 (plus personal research).
Before the later Neviim, ע and ח each had two pronunciations, like ש still does. And just as with ש, their secondary sound was more forward.
For ע, the main sound was as just I described, and the one still in use today. The secondary one was more forward, like Modern-Hebrew Resh. This ancient aspect of Hebrew is reflected in Ancient Greek and Latin transcriptions.
(It should be noted that some Mizrahi Jews pronounce ע as a pharyngealized glottal stop, which is like an א followed by ע. This is a modern pronunciation influenced by pre-modern colloquial Arabic and is different from the original pronunciation of ע. One proof is the above paragraph, which would not be possible if ע were a glottal stop.)