How to write 한국어

한글 consists of 19 consonants, 8 vowels, and 13 diphthongs (combinations of 2 vowel sounds).

The consonants are as follows: ㄱ(k), ㅋ(k'),ㄲ(kk), ㄴ(n), ㄷ(t), ㅌ(t'), 
ㄸ(tt), ㄹ(r), ㅁ(m), ㅂ(p), ㅍ(p'), ㅃ(pp), ㅅ(s), ㅆ(ss),ㅇ(ng), 
ㅈ(ch), ㅉ(jj), ㅊ(ch'), and ㅎ(h). 

For the most part, the consonants are pronounced exactly the same as their English counterparts. Of the consonants, there are 4 that have 3 types of variations: plain (p), aspirated (p'), and tense (pp); and 1 consonant has a 2-way contrast: plain (s), and tense (ss). The aspirated consonants are pronounced with a puff of air that is released from the mouth, while the tense can be described as requiring muscle tension upon release.

The vowels are: ㅏ(a), ㅐ(ae), ㅓ(eo), ㅔ(e), ㅗ(o), ㅜ(u), ㅡ(eu), and ㅣ(i).  
(a) is pronounced like the a in father, (ae) like the a in care, (eo) like the a in awake, (e) like the e in met, (o) like the o in owner (without the w sound), (u) like the oo in boo (but shorter), (eu) like the u in put, and (i) like the ee in see.

The diphthongs are a combination of any vowel sound and either the semivowel y- or w-:
ㅑ(ya), ㅒ(yae), ㅕ(yeo), ㅛ(yo), ㅠ(yu), ㅖ(ye), ㅘ(wa), ㅙ(wae), 
ㅚ(we), ㅝ(weo), ㅞ(we), ㅟ(wi), and ㅢ(ui).

Both ㅚ and ㅞ are both pronounced indistinguishably as (we), although they used to be pronounced distinctly. The diphthong ㅢ (ui) is pronounced as either (eu) or (ui) at the beginning of a word, or as (i) after a consonant or in the middle of a word.

Below is a summarization of the sounds in Korean:

한글 words are made up of single syllables, each having a consonant (or null) onset placed on the left, a vowel nucleus either to the right or below, and an optional consonant coda on the bottom. By null, I mean a silent consonant placeholder, represented by the letter ㅇ. Please take note that ㅇ also represents the sound (ng), but when it begins a word, it becomes silent! 

So when looking at the word 한글, you can see 3 parts to each of these syllables. The first syllable consists of the onset h- ㅎ, the nucleus -a- ㅏ, and the coda -n ㄴ. The second syllable has the onset g- ㄱ, nucleus -eu- ㅡ and the coda -l ㄹ.* So putting them all together it becomes 한글.

*There are certain instances in which letters change their sound when found next to other letters, 

In the case of the syllable not beginning with a consonant, such as in (eum), the result is 음, where ㅇ is the silent placeholder.

As seen in the last example, vowels appear in one of two places: to the right of the consonant, or beneath it. (a), (ae), (i), (e), and (eo) all are placed to the right, and (o), (u),  and (eu) are beneath. The diphthongs are a combination of both: 궤 (gwae) , 과 (gwa) , 귀(gwi).

Finally, there are a very few select consonants that can appear next to each other at the end of a syllable. These are: ㄱ+ㅅ, ㄴ+ㅈ,ㄴ+ㅎ,ㄹ+ㄱ,ㄹ+ㅁ,ㄹ+ㅂ,ㄹ+ㅌ,ㄹ+ㅍ,ㄹ+ㅎ, and ㅂ+ㅅ, as in, for example, 굶,값, and 많.

Besides these select consonant clusters, no other consonants may be found grouped together in the coda position of a syllable.

To make things simpler when learning to write Korean, you can think of each syllable of a block that is made up of 2 to 4 smaller blocks within. 

As a visual example, I'll provide an image from the Integrated Korean textbook.

In the case of a syllable of only a consonant and a vowel, there are two options (C represents a consonant and V a vowel):

p. 30

 or

p. 30

This can be seen in the examples 카 (kka) and 소 (so).

When there is only one consonant and a diphthong, you have:

p. 30

As in 궤 (gwe) and 돼 (dwae).

When the syllable is two consonants between a vowel,  or a vowel between a consonant and a consonant cluster, or two consonants between a diphthong, the following is the result:

p. 31

As in 김 (kim), 깎 (kkakk), 굶 (gulm), and 흰 (huin).

As you can see, each letter fills its block and each block is equally sized with one another, so that the resulting syllable is evenly spaced. It will take some practice to get this down, but you can do it!