Vowels are defined in a very simple way: height, backness, and rounding.
I should add that you can have vowels of different length (long or short vowels, literally, as in Latin, but not as the term is used for English-- those were historically long vs. short but now just different sounds, also why English differs from Latin pronunciation for vowels-- see https://en.wikipedia.org/wiki/Great_Vowel_Shift). And also that in some languages vowels can be nasalized (as in French) but (although often but not always there are fewer different nasal vowels than non-nasal, oral vowels) those are the same vowels, just plus or minus air going through the nose.Rounding is the easiest to understand: your lips can be rounded (as in Latin o, u) or not (as in Latin a, e, i). Back vowels tend to be rounded and front vowels unrounded, but some languages use rounding contrastively (various European languages have front rounded vowels, and Japanese has a back unrounded vowel like 'u' but unrounded, for example).
Height and backness refer to tongue position. For 'i' the tip of your tongue is very high and near the front of the 'vowel space' in your mouth. And for 'a' your tongue is back and low. It's very easy to diagram this, as in the IPA chart itself. That's actually a sketch of the mouth. (The consonant chart is similar, although not quite as literally a drawing of the mouth because manner is not really a physical dimension in the mouth although place is.)
More technically, and this is where it gets tricky, vowels are defined by their acoustic signature, which involves formants (
https://en.wikipedia.org/wiki/Formant). I won't try to explain the details here, but essentially there are bands of pressure/sound at different frequencies, and the frequency of those formants determines which vowel is being pronounced. The formants correspond, interestingly enough, to tongue height and backness. If you move your tongue up or down, or forward or back, you get slight variation in the first two formants (lowest exaggerated bands of energy in the acoustic signature of the vowel).
So what does all of that mean? Well, vowels are all almost the same thing (setting aside rounding), except for slight adjustments in the formants, e.g., tongue position.
What is the goal of the IPA? To represent contrastive sounds-- phonemes-- in the languages of the world. Some languages have very few vowel contrasts (Arabic has 3 for example). Others have many (like English with about 14 depending on dialect). What this means is that the IPA does not represent "sounds" but instead
phonemes, based on the grammar of a particular language. Phonemes don't make sense outside of the analysis of a particular language (you can't even compare them across languages in a very meaningful way), because they are defined by the contrasts they make within a language. So the IPA gives you some symbols to write down the different vowel phonemes of various languages, in fact enough symbols to cover the various height and backness contrasts in all of the world's languages. But that doesn't represent all of the different "sounds" we can make, nor do those symbol-categories have any real special meaning. Which symbol you use for a phoneme isn't very important (it's the contrasts that are!), and in fact any phoneme will represent a range of articulation probably covering several IPA symbols. Of course we usually use the "center" of a vowel's pronunciation as the symbol, and that's a good habit. But it's not very theoretically meaningful. For example, in Latin, the vowel /i/ (spelled "i") corresponds to a very similar sound (phoneme) in English: /i/ (as in "eat"); but that same Latin vowel phoneme /i/
also sometimes corresponds to the English phoneme /ɪ/ (as in "it"), because English makes the distinction between those two sounds but not Latin.
The reason that listening to IPA symbols isn't giving you a sense of "what the vowels are" is because your ear is trained for English, and without a lot of practice with other languages (Latin won't help much because (1) it has a subset of the vowels you already know in English, and (2) you're mostly exposed in writing not speech I assume so you won't be retraining your perception much anyway), you really won't be able to hear a difference in additional IPA contrasts. That's because they represent contrasts in languages you don't speak. To you, the difference between those sounds will just seem like slight variation within the same sound (if you can even hear/notice the difference at all). Once you cross an English phoneme boundary, the difference will be obvious. That's not accident: you've been practicing that your entire life as an English native speaker, and it's the same reason non-native speakers of English struggle with it. That limited/specialized perception is part of being fluent/native in a language. It just happens to make it harder to hear other languages, though. (There is some interesting research showing that infants at about 6 months old can hear all or most of the contrasts in any language, before they start to specialize in their native language.)
So how does this answer your question? Well, the answer is that you can't really just "look up what the vowels mean" because they simply refer to acoustic patterns with very fine levels of detail. Those contrasts are only meaningful (and "hearable") within a language where they are, well, contrastive. Trying to figure out what "all of the vowels are" is a pointless exercise outside of that. The IPA only makes sense because it's based on all of the languages studied to far by linguists (it's updated every few years!) with enough symbols to represent the contrasts found anywhere, not just in a single language. So it's like all of the sounds around the world were flatted onto a single chart based only on their 'central' pronunciation. That's what the vowel chart is. And more importantly, that's
only the contrasts found in known languages-- infinite variation is possible because those are just continuums (height and backness!) and you could make more and more (artificial) contrasts and corresponding symbols, if you wished.
Now, you didn't specify in the question whether you were talking about all IPA vowel sounds, or just the sounds of English and/or Latin. All of the IPA vowel sounds? Very difficult, see above. And for that it probably would be best to understand phonetics better than I can explain here (see the link about formants above, even better take a Phonetics class).
Latin is the simplest, because there are just 10 vowels: 5 long, 5 short, without any further contrast between those. So there are just 5 vowel symbols to worry about. Latin's "cardinal vowels" as they are called are 'spelled' the same way in IPA as in Latin (intentionally), so there's nothing to learn there. Done.
English is a little trickier. There are around 14 vowel phonemes in English, depending on your dialect. They
will be different IPA symbols in different dialects, so this is important. Wikipedia has several pages on "IPA for English dialects" or something like that so you can look up more there or pick your favorite accent to compare (try American vs. British vs. Australian to start, and then of course you can look at variations within those dialects).
English spelling is so far from 'phonetic' that it's a mess to compare to IPA. (I'm an experience linguist who reads IPA very often, and IPA-spelled English is still very confusing to my eyes, because English orthography seems like such an inherent part of English, even though it's actually a bizarre way to spell it, see above about the Great Vowel Shift.)
The easiest way to do that is to write out a list of words and their corresponding vowel symbol in IPA, at least to get started. I can do this for you for American English (roughly "General American"), but the details will vary by dialect! So you can try out your own too. The best way to do this is to pick a particular pair of consonants that actually have words with all or most of the vowels. One good option is "b_t" which can represent most sounds:
beat /i/
bit /ɪ/
bait /e/
bet /ɛ/
bat /æ/
bot /ɑ/
bought /ɔ/*
but /ʌ/
book /ʊ/
boat /o/
boot /u/
(A schwa /ə/ is of course found in any fully reduced vowel pronunciation in English, so you can find that simply by saying "but" very fast as you would when speaking quickly, something like 'but I don't want to' as 'but I don' wanna' /bədaidonwana/.)
(*By the way, a common example of a dialectal difference in American English is that only about half of Americans differentiate between "bot" and "bought", more commonly exemplified as "cot" vs. "caught", also known as the 'low back merger' because for many Americans those vowels have merged. And that's another reason IPA isn't convenient for replacing English spelling-- so many variations across dialects it sort of makes sense to keep what we have because finding one system for all dialects would be difficult and still require memorizing spelling for sounds we don't have as contrastive in our native dialects.)You can also add diphthongs to that list above, which are combinations of two vowels (so the formants are moving during the pronunciation, but you can think of them as literally combining the two IPA symbols that represent them). Some common diphthongs in English can be represented as follows (with only vowel usually being reduced to a consonant-like glide):
bite /ɑɪ̆/
bout /aʊ̆/
Boit [name] /oɪ̆/
butte /ɪ̆u/
etc.
(I should also add that many linguists represent the so-called 'long vowels' of English as diphthongs ending in a small glide or short vowel, e.g. /o/ as /oʊ̆/ or /e/ as /eɪ̆/ but I'm setting that aside here because it's not contrastive in English (so it is consistently predicted for any pronunciation and therefore doesn't need to be written) and keeps the notation simple.)In summary, vowels are very simple in a technical sense, but very unintuitive if your starting place is English spelling. IPA is a set of symbols we can use to represent contrasts in individual languages, but the contrasts are not symbols: they are relationships between symbols, and each vowel "sound" (phoneme) in a given language can probably be pronounced as several different IPA symbols. Vowels are not points in the IPA chart but ranges.
Listening to the IPA vowels (there's a good website from UCLA you can check out, with all of the sounds) is about the only way to try to understand this without getting into the details of articulation (as explained above). But as you found, your ears are not trained to hear the differences not found in English, so that isn't too much help either.
A very good website for English specifically (also Spanish and German, and now Chinese too) is offered by the University of Iowa:
http://soundsofspeech.uiowa.edu/index.html#englishClick one of the languages at the top, and then scroll down to the link to the 'website' version' (or download the app if you wish).
Once you launch that, you'll be able to explore the English vowel sounds represented by IPA symbols (for American English). What is extremely helpful here is that aside from being able to listen to the pronunciations, there are animations of the tongue for you to understand how the vowels are pronounced. I highly recommend this website for Phonetics students as well as for English learners.
As a bonus, Spanish is also available there, and Spanish has very similar vowels to Latin, so you can check that out too.
