Simply said calligraphy is an artistic way of writing japanese characters. Japanese language has three alphabets – hiragana, katakana and kanji. First two are simple syllabic alphabets, that have evolved from the kanji. The last one are characters that had been imported from China, and these are the one mainly used for calligraphy.
It is discipline very complicated. To what degree is often unimaginable for people who have never had a brush in their hands. The order of the strokes is very important, the character has to have right proportions and overall balance, and to top it all there is a philosophic overlap – the characters have several readings (basically sino-japanese and japanese, not seldom several from each one), and these often vary depending on what other characters they are being written with.
Moreover there are several styles, basic (but not only ones) being the kaisho, gyóshó and sóshó. Same characters are being written differently in each and every one of them, and sometimes the differences are quite severe. Therefore the study of calligraphy is a lifelong journey, and to be able to trully savour the mastership one has to develop feeling for the littlest of details.