Learn To Read Korean -- Part One
This is Part One in a multipart series of articles covering the basics of reading and writing in Korean. By the end, you should be merrily sounding out anything you run across 
(and doing it with better pronunciation than most foreigners I've met who've been here for years).
Korean is a very different language, structurally, from English and many European languages. For Korean students of English, and for speakers of other languages trying to learn Korean, it's a hard slog getting beyond the basics. To my continuing shame, although I can read and write the language with some facility, after nearly 10 years of exposure to it (and, I'll admit, study of it that has been at best haphazard and desultory), I'm very far indeed from fluency.
The good news, though, is that reading it is literally a snap. A few hours with the basics, and almost anyone can be up and running. Or walking, at least. The writing system is about 14,000 times simpler to learn (scientifically speaking!) than Chinese or Japanese, and truly elegant in its design, philosophy, and suitability for capturing the sounds of the spoken language.
Before we begin with the basics, you'll need to be able to actually see the Korean text in this page. Tutorials on how to install East Asian fonts (if you don't have them already) can be found at Wikipedia, for a variety of common operating systems.
If you can't see this -- 안녕하세요! -- then go and install the fonts, and come back. It's OK, I'll wait.
History
Right, let's begin with some background.
King Sejong was the 4th King of the Choson Dynasty. In 1446 (dates vary, as do details of the story), scholars of the government office chip'yon'jon, or the Pavilion of the Assembly of Sages, were appointed by the King to invent a new writing system for Korean. Until that time, Chinese characters had been used to represent the sounds of the syllables of spoken Korean (The characters are called hanja, and still sometimes used to this day in print. Learning a basic set of 1800 of them was until recently a compulsory part of the education of all South Koreans, and they still play an important part in place names and personal names).
Writing had for centuries been the province of the educated elites, and this new system (although scorned in early days as writing for 'women and children') was created with the aim of spreading literacy. It was a success -- Korea now has a literacy rate of 97.9 percent, one of the highest in Asia.
A book of instruction for the new writing system was published, called Hunmin Chongum: "The proper Sounds for the Instruction of the People". The script it introduced later became known as 한글 (in the new romanization, hangeul).
If there is sound natural to Heaven and Earth, then there should certainly be writing natural to Heaven and Earth. Thus the men of antiquity relied on sounds and designed characters, thereby to convey the circumstances of the Myriad Things and to register the Way of the Three Germinants, we of later generations cannot change them. However, the winds and soils of the Four Quarters diverge, one from the other and sounds and breaths, following them, are likewise different. Presumably because the outer kingdoms have their sounds but lack characters for them, they have borrowed the characters of Chinese to take care of their needs. This has been like a handle that ill fits the hole; how could they have been applied with out obstructions? -Hunmin Chongum
The Vowels
Besides its simplicity and elegance, one of the most fascinating things about the Korean alphabet is its grounding in the philosophical principles of the time, and its deliberate connections to the physical configurations of the organs of speech.
There are ten vowels (and eleven dipthong vowel combinations) and fourteen consonants (and five doubled consonants) for a total of 40 phonemes. Characters are shaped with symbols (dots and circles, horizontal lines, and vertical lines) that represent the fundamental elements of the cosmology: respectively heaven, earth and humanity. Heaven is a round dot, Earth is a horizontal line and the symbol of mankind is a vertical line. All the vowels in the Korean language are combinations of dots, horizontal and vertical lines. These signs are further balanced into the the opposing energies of yang (bright) sounds and yin (dark) sounds.
Here are the vowels:
The ten basic vowels are those with only one long straight vertical or horizontal line (earth and human): ㅏ and ㅑ, ㅗand ㅛ, ㅓand ㅕ, ㅜ and ㅠ, ㅣ and ㅡ.
If the dot (represented in this font as a shorter perpendicular line) is to the right of the vertical, we get some of the "bright" vowels: ㅏ ㅑ ㅐ ㅒ. If it's above the horizontal, we get the last two brights: ㅗ ㅛ.
If the dot (represented in this font as a shorter perpendicular line) is to the left of the vertical line, we get the "dark" vowels: ㅓㅕ ㅔ ㅖ. If it's below the horizontal, we get the other two darks: ㅜ ㅠ.
If there is no dot, the vowel is neutral:ㅣ and ㅡ
Adding a second dot (short perpendicular) to a vowel adds a "y" before the basic vowel sound("ah" becomes "yah", for example): ㅑ, ㅒ, ㅕ, ㅖ, ㅛ and ㅠ.
A horizontal vowel (ㅗ or ㅜ) can be paired with a vertical vowel to form a dipthong. The horizontal vowel always comes first in the pairing, and this results in a "w-" sound in front of the pure vowel to give us sounds like "wah," "weh," "wi," and so on: ㅘ ㅙ ㅚ ㅝ ㅞ ㅟ and ㅢ.
So far, we haven't matched any of the characters to their actual sounds, so don't worry if it's not coming together for you yet.
For that, I'm going to give you this link for basic vowels, and this one for dipthongs. Open it in a new tab or window, and mouse-over to listen to the vowel sounds as you look over what I've said about the vowels. If you repeat the sounds, think about the shape of your mouth as you make them, and how that relates to the bright/dark/neutral labels.
Coming Soon
If you want to skip ahead and listen to the consonants as well on those pages, they will be the focus of Part 2, where we'll see how the design of the consonants (which are created in all languages by the modification and interruption of the flow of air by the physical parts of the mouth) are cleverly modelled on the physical movements needed to create them.
Part 3 will deal with how syllables and words are put together, the relatively simple rules for reading and writing them, and the few but consistent exceptions.
Part 4 will talk about the difficulties and challenges of the romanization of Korean, why it's such a mess, why Koreans have so much trouble with English pronunciation (though they need not) and what you can do to make the situation better as a teacher (if you are one).
For now, one parting piece of essential advice to keep in mind: unlike English, the sounds of Korean vowels are (essentially) immutable. No matter where they are in a syllable, they make the same sound. This is one of the pure joys of learning to read Korean, and something that many (if not most) new learners of Korean miss, in part because of the confusion that reigns in romanization.
Have fun. You're about a third of the way to being able to read Korean!