Anime is the Japanese shortened word for 'animation'.
In Japan, 'anime' would be any animated production, from Disney to Legend of the Overfiend. Anything that is animated, be it Japanese, English or French. Outside Japan, though, 'anime' tends to refer, specifically, to Japanese animation.
In Japan, anime is a rather popular form of entertainment. It comes in the form of tv shows, movies at the cinema as well as direct to video releases (OVA - Original Video Animation). Unlike in the west, anime has a huge variety of genres - it is not only for children. Nor, on the other hand, is it just sex and violence for adults! Anime is for everybody!

Most of the time, anime comes from a manga - Japanese comics. Occasionally, though, you'll get an anime first, and the manga will follow... but as a general rule, anime comes from a popular manga series. The manga are sold in anthologies, either weekly or monthly, and have a huge number of readers! Even more, if you consider that these anthologies are read once, and left on the train (often by businessmen and office ladies on their way to work - manga, you see, doesn't have the same stigma as comics tend to have in the west) for someone else to read. (They are cheap throw-away items - the Japanese version of graphic novels follow, and these are more likely to be collected.) As with anime, manga comes in many genres for all age groups - manga for businessmen, women, boys, girls, romance, fashion, adult titles, sports and more. It's a huge industry in Japan, and very, very popular.
Rock music, sometimes also known as "rock and roll" is a style of music that became popular in the 1950s in America and Europe. Rock music is primarily based on older musical styles, such as the rhythm and blues music originated by African American performers such as Chuck Berry and Little Richard, with a heavy focus on guitar, drums, and powerful vocals.
To the uninitiated, any loud music is called heavy metal. In reality there are a multitude of heavy metal styles and subgenres. Heavy metal is a wide umbrella characterizing a style of music that is generally loud and aggressive. There are genres that are very melodic and mainstream, and other genres that are extreme and underground.
J-Rock
The modern Japanese music scene includes a wide array of performers in distinct styles both traditional and modern, ranging from rock, electro, punk, folk, metal, reggae, salsa, and tango to country music and hip hop. The word for music in Japanese is 音楽 (ongaku), combining the kanji 音 ("on" sound) with the kanji 楽 ("gaku" fun, comfort). Local music often appears at karaoke venues, which is on lease from the record labels.
The old Japanese music has no specific beat, and is calm. The music is improvised most of the time. In 1873, a British traveler claimed that Japanese music, "exasperate[s] beyond all endurance the European breast.
J-Pop
J-pop (ジェイポッ) is an abbreviation for Japanese pop, but is also a loosely defined musical genre that entered the musical mainstream of Japan in the 1990s. It refers to Japanese popular music, and was coined by the Japanese media to distinguish Japanese music from foreign music. Today, the Japanese music industry is the second largest behind the US in the world.