四字熟語 (よじじゅくご) or yojijukugo are what are known as "four-character compounds" or "four-character idioms" in Japanese.
These contain four kanji, and the meanings of the individual kanji don't usually have much to do with the meaning of the idiom as a whole. But the meaning of the idiom often comes from a story which explains the derivation of the four characters used.
Like the rest of kanji, yojijukugo come from China. Yojijukugo come specifically from the Chinese Chengyu. However, there are many Japanese-only yojijukugo as well. Each yojijukugo represent the moral of a Chinese story, a Japanese proverb, Buddhist scripture, or an old Japanese story. Therefore to really understand the yojijukugo meaning, you would need to know the story behind it.
Estimates of the number of existing yojijukugo range upwards to 20,000 which is quite a lot of stories to learn. However, some are fairly straightforward and don't require a lot of information to understand the meaning. You can learn some of the most common ones without having to learn every story behind them.
一期一会 いちごいちあい
Once-in-a-lifetime encounter
一石二鳥 いっせきにちょう
Kill two birds with one stone
一生懸命 いっしょうけんめい
With utmost effort
温故知新 おんこちしん
Learning from the past
以心伝心 いしんでんしん
Without the need for words
花鳥風月 かちょうふうげつ
The beauties of nature
初志貫徹 しょしかんてつ
Achieving one's original intention
質実剛健 しつじつごうけん
Unaffected and sincere
十人十色 じゅうにんといろ
To each his own
清浄潔白 せいじょうけっぱく
Pure in heart and with a clean conscience
天真爛漫 てんしんらんまん
Simple and innocent
日進月歩 にっしんげっぽ
Steady progress
臨機応変 りんきおうへん
Playing it by ear
These are just a few of the four-compound idioms that Japanese has to offer. What is your favorite yojijukugo?