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A quick refresher - Kanji are nothing like the friendly Hiragana and Katakana you've encountered so far, and they're definitely nothing like the English alphabet. Think for a second - what images pop into your head when you think of Chinese or Japanese? If they look like this...

You'd be right!
Origins
Before we go on to learning about Kanji in Japanese, we first need to understand where they came from and why they mean anything.
The beginnings of Kanji go back to ancient China, where legend has it that a man named Cangjie created Kanji based on animal footprints. Nobody really knows the real deal, but it's simple enough to say that Kanji were originally drawings of things, like cavemen depictions in caves, except they were found on bones and shells (for finding the will of the gods).

Shown above, are the following characters in Oracle Bone Script: From left, 馬 ‘horse’, 虎 ‘tiger’, 豕 ‘swine’, 犬 ‘dog’, 鼠 ‘rat and mouse’, 象 ‘elephant’, 豸 ‘beasts of prey’, 龜 ‘turtle’, 爿 ‘bed’, 為 ‘do or for’, and 疾 ‘illness.’
Now, note that I said "Kanji were originally drawings of things." This is still true today, but the system is a bit more organized and simplified to be aesthetically pleasing (good on the eyes, in other words). Some parts of Kanji are a bit more abstract than their original shapes. Below is an early form of stylized Kanji, written in what's called Bronze Script.

Looking at the first picture above, you may not understand the "system" behind the Kanji at all - they all look like a collection of random strokes. In another lesson we will learn about the composition of Kanji and the meanings of the different parts (radicals, as they're called) of them, but for now just understand that Kanji are not random strokes.
Kanji in Japanese
For a long time, the Japanese language did not have a written form. After the Korean language implimented Kanji into their language, the Japanese did the same as they began communicating with each other a few hundred years ago. Originally, the Kanji were used only for the phonetic value. That is, each Kanji had a pronunciation that would match a sound in Hiragana or Katakana (plus a few more older sounds that are no longer used), but the meanings of the Kanji were ignored. In fact, this is exactly where Hiragana and Katakana came into play.

* ゐ, pronounced "wi," no longer used in modern Japanese.
When women began learning how to write Kanji, they stylized them in a cursive form. This cursive style, simplified over many years, brought out Hiragana. On the other hand, men took out small parts from the same Kanji (or others similarly used) and brought about Katakana. They are still used as syllabaries today, as you learned in the first two lessons.
With Hiragana and Katakana taking on a phonetic role, Kanji were no longer used for just their sound. So you're probably wondering, what did they do with them? Simply put, they imported the Chinese pronunciations of the Kanji into the Japanese language, and at the same time applied their native Japanese words to Kanji with similar meaning. In this way, Kanji have two types of pronunciations.
On'yomi
Since the Japanese took the Chinese pronunciation of Kanji and changed them to the phonetics of the Japanese language, Kanji in Japanese have a "pronunciation" (so to speak) that is called "on'yomi." Kanji tend to have at least one on'yomi (although they can have more than one, or none at all).
On'yomi are most frequently used in the pronunciation of words with more than one Kanji in them, as these words tend to be imported vocabulary from Chinese. Quite rarely though, on'yomi can be used in words that have only one Kanji.
In a dictionary, a Kanji's on'yomi are written in Katakana.
Kun'yomi
The Japanese also applied their native vocabulary to the same Kanji. This pronunciation is called "kun'yomi." In general Kanji have at least one Kun'yomi, but like On'yomi, there can be more than one or none at all.
Kun'yomi are mostly used in words that contain only one Kanji, and have trailing Hiragana that is a part of the word with it. Occasionally you will see kun'yomi in a word with more than one Kanji in it. In the rarest occasion, you can even see a mixed-reading compound (one Kanji uses its on'yomi, the other uses its kun'yomi).
In a dictionary, a Kanji's kun'yomi are written in Hiragana.
Let's look at an example. The Kanji 「食」has the on'yomi 「ショク」, and the kun'yomi 「た・べる」. It uses its on'yomi in the word 食事 (ショクジ, a meal) but its kun'yomi in the word 食べる (た・べる, to eat).
Pronunciation
In general though, it's not always easy to predict the reading of a Kanji based on the type of word it's in. There are some cases where a lone Kanji uses its on'yomi, and cases where a word with multiple Kanji may use kun'yomi. As a general rule, it is best to learn the pronunciation of the word and focus less on the pronunciations of the Kanji themselves.
Uses of Kanji
You may wonder though, since there are Hiragana and Katakana, why the Japanese even needs to use Kanji at all. After all, Hiragana and Katakana are simpler and represent all the sounds neatly. Here's the thing - the Japanese language, due to its small set of sounds, has lots of homophones. And I mean lots.
If you look at this dictionary search for 「だん」 you can clearly see that there are many kanji that share that same reading that all mean different things. Hiragana alone isn't able to differentiate between these meanings, and that's why we have Kanji - to represent meaning.
Learning Kanji
Most textbooks teach you Kanji in the order that the standardized tests list Kanji (essentially the most common first), which misleads students into thinking that Kanji are completely random and scary. The level of a Kanji is not determined by stroke count, but rather what if it has a simple meaning and how often it's used, which in turn affects when it is taught in books. Essentially, Kanji like 「飲」 (to drink) show up before ones like 「寸」 (an "inch")!
At the same time, they have you write Kanji over and over to "help you remember them," but the reality is you will completely forget the Kanji a few minutes later. This guide aims to teach Kanji in a systematic manner, by starting with the small parts of Kanji (the radicals) and building up to make more complicated Kanji. In other words, you will be able to learn new Kanji with ease by learning the basic parts of all Kanji first.
In this way, Kanji will hopefully become easy and maybe even enjoyable to learn. Learning them using their radicals becomes almost like solving a puzzle - you put the pieces together and see the whole picture, and then it clicks.
Images taken from Wikipedia.
Lessons
The Writing Systems
- Understanding the Japanese Writing Systems
- Hiragana
- Katakana
- Romaji & Why It Sucks
- Introduction to Kanji