Korean Vowels

There are 21 Korean vowels, but they can be seen as combinations of only 6. Those 6 can be broken down to two dimensions: strong/weak/neutral and horizontal/vertical. All the strokes are straight and are either long or short.

Let's start with neutral horizontal: ㅡ
If you have an expressionless face, open your mouth a little and make a sound, it's basically this. You can think of it as an "oo" sound without rounding the lips, or you can think of it as an "mmm" sound or hum with the lips slightly apart.

Neutral vertical: ㅣ
This is an "ee" sound.

For the strong vowels, add a short stroke out and up:
ㅗ ("oh")
ㅏ ("ah")
(I introduced these vowels last time.)

For the weak vowels, down and in.
ㅜ ("ooh")
ㅓ ("uh")

That's it for the basics as I see them. See if you can remember their sounds:
ㅡ ㅣ neutral
ㅗ ㅏ strong
ㅜ ㅓ weak

Now consider just the verticals and we'll add two more: ㅐ and ㅔ. Visually, you can think of them as ㅏ + ㅣ and ㅓ + ㅣ. So they are strong and weak, respectively. ㅐ is pronounced "ay" as in hay and ㅔ is pronounced "eh" as in bed. ㅐ is more open than ㅔ; if you make the sounds for "ㅣ, ㅔ, ㅐ, ㅏ" in that order you'll know what I mean. Try it.

Now consider the strong and weak vowels:
ㅗ ㅏ ㅐ (strong)
ㅜ ㅓ ㅔ (weak)
Each of these are simple sounds, monothongs, which means that the mouth shape doesn't change while saying them. But they each have a corresponding diphthong that begins with the vertical neutral sound ("ee"). The resulting symbols just add another short stroke.
ㅣ + ㅏ = ㅑ
ㅣ + ㅐ = ㅒ
ㅣ + ㅓ = ㅕ
ㅣ + ㅔ = ㅖ
ㅣ + ㅗ = ㅛ
ㅣ + ㅜ = ㅠ
What do these symbols sound like? Let's take ㅑ. You start with ㅣ ("ee") and end with ㅏ ("ah"). Say it more and more quickly and you get what sounds like "yah". Same goes for the others; the vowels with two short strokes sound like the corresponding one-short-stroke vowels with a "y" added before it. But keep in mind that Koreans don't think of "yah" as a consonant sound plus a vowel sound (y + ah) but as a single vowel sound ㅑ. In the same way, we think of the English sound "I" as a single vowel sound even though it is a diphthong made of ㅏ and ㅣ, "ah-ee" said really quickly (this is more obvious to formal singers who have to hold out syllables more often).

On to combined vowels. The strong horizontal vowel (ㅗ) can be combined with the strong vertical vowels (ㅏ & ㅐ); likewise for the weak horizontal vowel (ㅜ) with the weak vertical vowels (ㅓ & ㅔ). These make diphthongs similar to the ones just above but with a 'w' sound rather than a 'y' sound.
ㅗ + ㅏ = ㅘ
ㅗ + ㅐ = ㅙ
ㅜ + ㅓ = ㅝ
ㅜ + ㅔ = ㅞ
Again, we'll take the one based on ㅏ as our example. To pronounce ㅘ, you start with "oh" and end with "ah". Say it quickly and you get something like "wah." Likewise with ㅙ: you start with "oh" and end with "ay" so you get "way." The next two are similar but because you start with "ooh" rather than "oh" the sound quality of the "w" sound is slightly different. It isn't very crucial to be able to distinguish these two to comprehend Korean speech. As one of my books puts it, the only difference between them is the "degree of clearness of the sound": clear ㅗ vs. dark ㅜ. I don't know how helpful that is, but you if you're determined to pronounce them exactly right, just make sure to make yourself think of each of these sounds not as a "w" plus a vowel sound but as a diphthong made from a particular pair of vowels. So for ㅝ, you start with "ooh" and end with "uh," getting "wuh"; and for ㅞ, start with "ooh" and end with "eh" to get a "weh" sound, but you think of this more in terms of a semi-vowel "ooh" than of a consonant sound "w".

All three of the horizontal vowels can combine with the neutral vertical vowel ㅣ, so there are three more combined vowels:
ㅜ + ㅣ = ㅟ
ㅗ + ㅣ = ㅚ
ㅡ + ㅣ = ㅢ
The first one here, ㅟ, is the most intuitive. You start with "ooh" and end with "ee", getting "wee". It seems like the next one, ㅚ, should sound extremely similar to this but in actual speech instead of ending with an "ee" sound, for some reason it actually sounds basically the same as ㅙ. As for the last vowel, ㅢ is just a combination of the sounds ㅡ and ㅣ. This has no equivalent in English but if you just say the two sounds together quickly, just like in the other diphthongs, you'll get the sound right.

There you go! All 21 vowels derived from the basic six. Here they are in their dictionary order:
ㅏ ㅐ ㅑ ㅒ (strong vertical)
ㅓ ㅔ ㅕ ㅖ (weak vertical)
ㅗ ㅘ ㅙ ㅚ ㅛ (strong horizontal/combined)
ㅜ ㅝ ㅞ ㅟ ㅠ (weak horizontal/combined)
ㅡ ㅣ ㅢ (neutral)

I'll end with a reminder that even though I presented the vowels by themselves in this post, they always appear with an initial consonant (and sometimes an ending consonant on the bottom as well) and never alone. Even when there is no consonant sound, they appear with a placeholder consonant, ㅇ. With the vertical vowels, the initial consonant goes to the left of the vowel; with the horizontal ones, the initial consonant goes above the vowel; with the combined vowels it goes above the horizontal part and to the left of the vertical part. So here are some examples using the familiar consonants ㄱ and ㅇ.

개 (dog)
여기 (here)
우유 (milk)
과거 (the past, days gone by)
귀 (ear)

Now you only need to learn the consonants to have a pretty solid base in written and phonetic Korean.