Shina (支那) is a Japanese word for "China," formerly in widespread usage, but now considered extremely derogatory and offensive by Chinese due to its associations with Japan's former attempts colonial domination over much of China. Today the much more politically correct word used by Japanese to describe China is Chugoku, which is the Japanese pronunciation of the Chinese word for themselves, Zhongguo or "Middle Kingdom."
Oddly enough, the term Shina was actually coined by the Chinese themselves, thousands of years ago, when Chinese Buddhist scholars rendered the Sanskrit word for China, Cin, into Chinese phonetically as 支那 (Zhīnà). This was only one of many words for China, but it was in widespread use throughout Classical Chinese literature for centuries, and by this means was imported to Japan.
In the Early Modern Period, when the Japanese learned that the Portuguese called China Sina, they figured this must be the "true" name for China, and began using the world much more than other terms. By the Meiji Period, well before Japanese colonialism, the term had already become firmly ensconced in the Japanese lexicon as a standard term for China in general, and particularly when translating western texts into Japanese, as a way to render the western word "China."
Meanwhile, the word continued to be used by Chinese themselves, well into the 1920s, although rising Chinese nationalism meant that the word increasingly came to be rejected as foreign in favor of calling China "The Middle Kingdom," with it's overtones of China being located at the center of the World.
By the height of Japanese imperialism in China during the second Sino-Japanese War of 1931-1945, however, the Japanese were the only people still using the kanji compound "Shina" to refer to China, so it came to be increasingly viewed as derogatory and offensive.
After World War II ended the Republic of China officially requested that the Japanese government cease using "Shina" to refer to China in official documents, and the Japanese government complied. Gradually the word almost totally disappeared from the Japanese language, until today, when you can hardly find it unless you go back to old prewar documents or extreme right-wing propaganda.
Still, you can occasionally find an extremely elderly people in Japan occasionally referring to China as "Shina" or Chinese people as "Shina-jin" in a slip of the tongue or a moment of forgetfulness. The word is also retained (in katakana form) in several geographical names such as Higashi Shina Kai ("East China Sea") and Indoshina ("Indochina").
In fact, Shina is such a forgotten term in Japanese that many Japanese people don't realize that it is considered offensive to Chinese, instead considering it a sort of quaint and old-fashioned term that old people use, like when old people in America use the word "Negro" to refer to blacks without actually intending any offense.
In China, however, where many young people were educated in the anti-Japanese education system in place in the 1980s and 90s, many young people are keenly aware of the term Shina and view it as an extremely derogatory ethnic slur. There is even a frequently repeated claim that the Japanese deliberately concocted the term "Shina" to slight Chinese, based on the fact that the first character of "Shina" means "branch," "subsidiary," or "supporting role." This claim, of course, completely ignores the fact that the character was chosen only for its phonetic value, and was actually chosen by the Chinese thousands of years ago.