The Japanese character that the Taka's younger son paints and gives to Algren is the character for "samurai".
The 2003 movie The Last Samurai tells the story of an American military advisor embraces the Samurai culture he was hired to destroy after he is captured in battle.
The movie starred Tom Cruise, Ken Watanabe and Billy Connolly. See full cast and crew at IMDb.
Japanese consists of two scripts (referred to as kana) called Hiragana and Katakana, which are two versions of the same set of sounds in the language. Hiragana and Katakana consist of a little less than 50 "letters", which are actually simplified Chinese characters adopted to form a phonetic script.
Chinese characters, called Kanji in Japanese, are also heavily used in the Japanese writing. Most of the words in the Japanese written language are written in Kanji (nouns, verbs, adjectives). There exists over 40,000 Kanji where about 2,000 represent over 95% of characters actually used in written text.
Learn more about Japanese writing system at Learn Japanese.