Although haiku are known for distilling profound meaning into a few syllables, 四字熟語 (4-character idioms) take this even further: 4 kanji to illustrate a truth, observe human nature, or tell a story. Before we get to 四字熟語, let's see which other ways kanji appear in written text.
Some kanji are quite fine by themselves, surrounded by kana or nothing at all, such as 夏 (summer); when I visited Hokkaido in August, this kanji was everywhere.
Many participate in 2-character compounds called 熟語, which means kanji compound in the linguistic sense, and idiom in the general sense. (More on idioms later.) These compounds, such as 美人, or 熟語 itself, are the workhorses of Japanese vocabulary, pairing kanji that supplement, contrast with, or reinforce one another.
For example, 価値, value, combines two kanji that both mean value or price, and both share a 訓読み: あたい; but I've seen 価値 in use more often than either of its parts alone. It seems to descend from the Chinese 價值 (or 价值 in simplified).
You'll eventually stumble across long strings of kanji, often businesses and government agencies, such as 海洋研究開発機構, meaning "Organization of Ocean Research and Development."
The most interesting way of combining kanji, in terms of culture, history, philosophy, and even humor, is the 四字熟語. 四字 is four characters; 熟語 is the same compound or idiom we saw earlier. 熟 itself means ripen or mature, and 熟語 brings to mind something cooked or baked that's more than just the sum of its ingredients: the difference between a cake and a pile of its raw ingredients.
On the surface, any four kanji could form a 四字熟語, such as 立入禁止 (do not enter), 高速道路 (freeway), or even the term 四字熟語 itself. These examples are literal and without much profound meaning.
Another level deep: idiomatic phrases whose symbolism is close to the surface. Example: 一石二鳥, which happens to map to the English phrase "two birds, one stone" on both the literal and idiomatic levels. Another example: 十人十色 (“10 people, 10 colors”) as "to each their own; different strokes for different folks". You can usually figure these out without a background in Japanese or Chinese history.
Going deeper brings out 四字熟語 that are impenetrable without context. For example: 海千山千. It's a mix of 訓読み and 音読み and literally means "sea thousand mountain thousand". So... long journey? Perserverance? Time?
The answer is "sly old dog" or "a crafty person who’s seen it all". That could arise from the idea of a person who's well traveled. But there's more to the story; background from gogen-allguide:
世の中で様々な経験を積み、物事の裏表を知り尽くしてずる賢いこと.
By accumulating (piling up) various world experiences, seeing completely all sides of things, one is crafty or cunning. Crafty, not simply wise; an interesting distinction. There's a legend behind this:
海に千年、山に住んだ蛇は竜になる (In a thousand years in the ocean, a mountain snake becomes a dragon; or: A snake must live a thousand years in the mountains and the ocean before becoming a dragon.)
From the above we have 海に千年山に千年, which is too long for a 四字熟語, so the "years" were taken out. One more wrinkle: 竜になる (becoming a dragon) has a positive connotation (立派, "splendid, fine"); for example, the actor Jackie Chan's stage name is 成龍 (become dragon). In order to preserve 竜になる for non-crafty people, it could not be used to praise the crafty ones.
To find out more, there are links below, as well as various books and articles. A Chibi Maruko-chan book, 四字熟語教室 ("Yojijukugo Classroom") (reviews) treats several examples in detail using manga.
There are also opportunities for 冗談 (jokes) and mischief: a slight alteration to 弱肉強食 (survival of the fittest; "the weak are meat the strong will eat") results in 焼肉定食 (set meal with grilled meat)!
Even though thousands of 四字熟語 exist, you can still make up your own. To lament my experience trying to remember kanji, I came up with 朝覚夜忘: learn in the morning, forget at night. It takes a few times forgetting before the remembering takes root. I haven't seen this elsewhere, so I'm feeling pretty good about it being original.
Another: 自分話和: "Speak by yourself and find peace." This was inspired by seeing a member of our conversation group working out a sentence softly by herself before speaking to the group. (It will take some time to get good at inventing 四字熟語.)
What kinds of 四字熟語 might you come up with?
As of September 2017, even telecom providers are seemingly getting into the 四字熟語 game, with print ads stating 一應倶全, touting some no doubt expensive plan that is "complete with everything." However, this example is a common Chinese idiom that has not been adopted in Japanese. (Source: 500 Common Chinese Idioms: An annotated Frequency Dictionary)