The term "kanji" has come into use in the English lexicon, but its implications are often not fully understood. "Kanji" is a Japanese word that refers to the component of the language borrowed from China. The other components are the Hiragana and Katakana syllabaries, as well as Roman letters and Arabic numerals.
While kanji are Chinese in origin, they are not necessarily consistent with modern Chinese. PRC (Mainland) China carried out a widescale simplification program under its government in the 1950's. Although only a few of the characters were changed, they were the most widely used and as a result Simplified Chinese characters are for the most part unintelligible to students of Japanese or Traditional characters.
In addition, Japan has carried out its own character simplifications, although being more subtle and limited compared to China's reforms. For this reason, kanji are sometimes incompatible with Traditional characters used in Taiwan and Korea. In addition, a very small number of Kanji are Japanese born, and exist only in Japan.
Kanji were adopted into the Japanese language using two methods. The first adopted the Chinese reading (i.e. the sounds) in addition to the characters that represented them. These readings are called the "On"(音) reading, or "Onyomi"(音読み). Since Japanese is not linguistically related to Chinese, these sounds are an approximation at best. The second method was to "attach" native Japanese words to the characters. These readings are called the "Kun"(訓) reading, or "Kunyomi"(訓読み). Since all Chinese characters represent a Chinese word, nearly all kanji have an "On" reading, but not all kanji have "Kun" readings.
Chinese, in all its simplistic glory, generally has only one reading per character. If only kanji were that simple. While it is common for a single kanji to have one of each of the readings mentioned above("On" and "Kun"), it can easily have more.
The adoption of Chinese characters into Japanese took place over centuries, and came in different waves by different groups. For this reason it is not uncommon for a kanji to have one or even two alternate "On" readings. These alternate readings often indicate a different realm of meaning as well. For example, the kanji 経 is most commonly read as "kei", and roughly means "passage", or "through". However it can also be read as "kyō", in which case it means "sutra". This is because the character came into Japanese through two different waves that took place at two different points in history, one by scholars and the other by Buddhist monks. Alternate readings/meanings can often fall within different semantic "realms" for this reason.
Although multiple "On" readings can be possible, multiple "Kun" readings are much more common. One cause of this is overlap in the meaning of native Japanese words which led to them being attached to the same kanji. In addition, Japanese, unlike Chinese, is an inflective language with lexical variants (like "reduce" and "reduction" in English). The most common example of this are Japanese transitive and intransitive verbs, such as "oriru"(to go down), and "orosu"(to let down). Variants like these generally share the base they have in common ("o" in the above example) as the reading for the kanji. The next section explains this in better detail.
Because native Japanese verbs and (some) adjectives inflect, they require a little more flexibility than kanji can provide. Okurigana is used for this purpose. "Okurigana" are Hiragana letters that trail the kanji and indicate its inflected meaning. For example, ながい ("nagai") means "long". Its negative form, "not long", is ながくない ("nagakunai"), its past tense, "was long", is ながかった ("nagakatta"), and it's past negative, "wasn't long", is ながくなかった ("nagakunakatta"). From this pattern, you can see that while the ending changes to indicate different inflected forms, the base remains unchanged and provides the basic root of meaning. For this reason, the base ("naga" here) is included as the reading of the kanji, while the ending ("i","kunai",etc.) is written out following the kanji as follows - 長い, 長くない, 長かった, 長くなかった. This combination of kanji and Hiragana makes it easy to pick out words, and it is largely for this reason that the Japanese written language does not require spaces. When a word appears in modern Japanese, the okurigana will be written out after the kanji, while the kanji itself is considered to stand in for the base.
The majority of kanji are composed of two elements, the "radical" and the phonetic element (see The Ideographic Myth below for more on the phonetic aspect of kanji). Knowing the radical of a kanji is beneficial in two ways — it hints at the meaning of the kanji, and it also allows you quickly look up the kanji in a dictionary. Radicals can appear anywhere, even surrounding a character, but are often found on the left side of the character. Also, the entire character itself can also be a radical.
Radicals evolved from stand-alone kanji, but their shape as radicals can be somewhat different. An example of a radical is 人, also a stand-alone character meaning "person". As a radical, however, it appears on the left in slightly different form in characters such as 体("body"), and lends its meaning of "person" to the meaning of the kanji. Unfortunately, like many radicals this isn't necessarily of much help, as "people" related concepts are fairly universal, and thus rather abstract. Radicals with more concrete meanings tend to be more helpful to determining the meaning of a particular kanji. However, they often provide a helpful device for remembering a character, and are an essential component of kanji.
Strokes and Stroke Order
"Strokes" refers to the number of brush strokes it takes to write a particular kanji. Although this is fairly straightforward, it is not always intuitive. For example, a right, then downward stroke forming a corner is counted (and written) as one stroke. It should look smooth, and the two lines should not show any separation. Any character with this component in it will count it as one stroke which is why, for example, 口("mouth") is considered as three strokes instead of four, and 日("sun") is considered as four, not five. Other exceptions exist, but are generally few in number.
Radical stroke numbers may also seem counterintuitive depending on the reference they show up in. A basic character like 心("heart") has four strokes. In its left-appearing radical form, as in the kanji 性("characteristic"), it has three strokes but is sometimes still counted as four. However, this only affects the number of strokes in the radical by itself (i.e. when looking the radical up in a dictionary), not the total number of strokes in the kanji, so the above example would still be eight strokes, not nine.
Stroke order is also an important consideration when learning and especially writing kanji. Characters can look odd if written using the wrong stroke order. Very roughly speaking, kanji are written left to right, top to bottom. However there are numerous exceptions to this rule. It is imporant to recognize different kanji components that break this rule — they will nearly always have the same stroke order when they appear in other kanji. By learning the patterns you will be able to anticipate these exceptions.
A good example is the character 国("country"). The "top" of the surrounding box is written first, then the inner component, then the bottom-most line that "closes" the box. This is a difficult rule, but once learned remains consistent, and all other characters like this (ex. 困,図,回,因, etc) are written in the same way.
Intuition about the placement and direction of strokes takes time, but it is essential to writing ledgible kanji. The best advice is to always pay attention to detail and never assume that a strange looking element is just a sloppy or stylized version of another. A good example of this are the characters 石("stone") and 右("right"). At first glance it is easy to assume that they are the same character, but in fact they have different meanings and even different radicals. It doesn't help that kanji are often stylized, with lines or dots extending and hooking in different ways. Only intuition can tell you which of these lines are essential to the character to avoid confusion.
The "level" of a kanji indicates its level as determined by the Japanese Ministry of Education. The kanji in the lower levels are generally, but not always, more frequently seen in modern Japanese. They also represent the levels of grade school in which they are taught, the sixth being taught in sixth grade. These first six levels are known as the 教育("kyōiku") kanji and are made up of 1006 characters. Above this are the 常用("jōyō") kanji, which total 939 characters. In total there are 1945 basic kanji that are considered the minimum level required to read basic Japanese.
Although Chinese characters originally carried only one meaning per character, as with any language meanings have diverged and evolved over the years. Nowhere is this more true than in their usage in Japan. In the dictionary, synonymous meanings indicate different nuances of the same meaning. However, it is not uncommon for a single kanji to appear with two or more meanings that are completely unrelated to each other. In these cases, the kanji by itself still carries its original meaning, but auxiliary meanings have become attached to the character when it appears within certain contexts. In the case of kanji, this means when they appear alongside other kanji within certain compounds. For example, the kanji 明 appears with the meaning "bright, clear, next". Here, when appearing as an adjective like in 明るい, it simply means bright. However, it carries its meaning of "clear" when occuring a compound like 明確 (precision). Likewise, it can mean "next" when occuring in a compound like 明日 (tomorrow). It should also be noted that when these auxiliary meanings occur, the others no longer apply.
The Ideographic Myth
One final aspect that is important to note is that kanji(and Chinese characters in general) are not ideographic — that is, they do not represent ideas, but words. This point is commonly overlooked, as such common characters as 山("mountain"), 川("river"), 目("eye"), etc. were created based on drawings of the things they represent. However, no functional Chinese character today is a direct representation of an idea, they are all representations of words and in that regard they are phonetic(like English) despite their complicated morphologies. Moreover, Chinese characters that were created in the way described above are in the vast minority, and more than two-thirds are made up of two components — a radical lending its meaning, and a phonetic lending its pronunciation.
Fortunately for students though, this means that the majority of characters provide a hint to their meaning and their pronunciation. For example, the character 申("to say") by itself has the pronunciation "shin", and lends that pronunciation component to the characters 伸("stretch"), 神("god"), and 紳("gentleman"), all pronounced "shin". We're not always this fortunate — many kanji are made up of components that are unique, or have unpredictable readings. But the rule often holds true, and generally knowing a component's pronunciation will provide at least a hint in the right direction.
For more information, pinyin.info hosts a wonderful, in-depth essay on the subject of Chinese characters as a phonetic writing system:
The Ideographic Myth