Kata, or forms, are formal exercises, a detailed pattern of movements usually performed solo or with a partner. While any fixed performance piece or drill (a monologue, a musical piece, etc.) may be considered a kata, the term kata is usually used in the martial context. Some simpler kata may look like a sequence of drills strung together, but most kata do not. A kata is a performance piece often used to transfer techniques, principles, style, and emphasis. Older kata were not written down until recent times and have almost certainly been modified during transmission. Like a musical piece, the same kata may be performed with significant differences between different styles, schools of the same style, and individuals of the same school. The essential "tune" is usually recognizable, but some times the "dialects" may have drifted by tremendous degrees.
Every language may or may not have a term equivalent to kata. Here
Japanese uses the word kata = 型 = 形 = form. The Japanese language has no singular or plural variations, hence 1 kata, 2 kata —like 1 deer, 2 deer.
The embusen of a kata is the rough pattern made on the floor by the kata.
A kiai is a spirited shout sometimes done at key points in a kata.
Bunkai are the direct applications or interpretations of the moves in kata. Oyo ~ semi-free are variations on applications or interpretations of the kata. Toridai and himitsu are oyo that are not obvious to the casual observer.
English uses words like forms, sets or plays, but the word kata is precise and unambiguous. The English language would do well to steal yet another foreign word!
Korean uses words hyung = 형 = 形, or poomsae = 품새, or tul.
Sokugi taikyoku sono ichi. Like Taikyoku 1, except kansetsu geri = stomping kicks to the knee are used added on turns and moves 6-8 and 14-16 are front kicks.
Sokugi taikyoku sono ni. Like Taikyoku 2, except yoko geri = side kicks are used on turns, and moves 6-8 and 14-16 are front kicks.
Sokugi taikyoku sono san. Like Taikyoku 3, except yoko geri = side kicks are used on turns, moves 2, 4, 10, and 12 are inside outward roundhouse kicks, and 6-8 and 14-16 are outside inward roundhouse kicks.