For our readers in Russia not well familiar with such dance styles as modern dance and jazz dance, we shall try to give a very brief explanation, basically compiled from materials on Internet (look Internet links for details).
Classical dance (ballet) can be told being the base of all scenic styles of dance. Besides the statement of dance as high art and the self-valuable kind of theatrical activity, capable to develop a plot with no help of singing or recitation, ballet has worked out a terminology of dance and system of dance training, used with some changes in other styles. Therefore the dancers are recommended to start from the lessons of classical dance, even if lately they've specialized in modern dance, show-ballet or jazz dance.
Modern dance appeared in North America in the beginning of the 20th century as a way to express new feelings and concepts of the art of that period in dance, and as a reaction to poor state of the Western traditional ballet of that time, before it's revitalization by great Russian ballet, not well known in Europe until Diaghilev's "Russian seasons".
The term "modern" is more used in the USA, the term "contemporary" is used more often in Europe. Sometimes a distinction is made -- a few large schools that formed this style are called "modern dance" and all the other styles are named "contemporary dance".
Modern dance denies many characteristics of ballet or considers them unessential -- turned out positions, elevation, extension of body parts and dance on the tips of the toes. Very often dancers are bare-foot and widely use a floor technique. There are many different schools in modern dance which have incredible differences in their techniques. The variety of expressive elements of modern dance makes it considerably more democratic and gives the opportunity to perform successfully to the people plastically gifted but started to dance being adult or having body conditions not sufficient for ballet.
On West classical ballet and modern dance develop in mutual influence. Many outstanding dancers are equally strong in both styles and many choreographers freely combine the characteristic plastic elements of one and another styles.
Jazz dance arose as American urban social dance, indissoluble connected with jazz music and evolving with it, absorbing the elements of different ethnic and European dances. In the period when the widest popularity of jazz music coincided with the growing statement of creative dance on the American scene on behalf of ballet and modern, jazz dance was developed as the same serious style of scenic dance, becoming a basis of many settings, for example the famous "West Side Story".
The characteristic feature of jazz dance is use of a body for music rhythm beating, like the musical tool or the whole drum machine. To follow the rhythm, the gravity center should be low, therefore jazz dance does not have such weightlessness and lengthened lines of the body as ballet does. Also, the step accent is not a push, but a setting of the foot down (tapping), the momentum is directed down to the ground. Isolation is the other typical technique when one part of a body (torso, pelvis, limb...) moves independently on the others and follows its own rhythm.
The golden age of jazz dance ended in the 60s, with arrival of new music and dance styles (rock&roll, break, disco...). Now only a few groups can show the true jazz dance and virtuous acrobatics of the others don't differ from the same acrobatics of contemporary groups.
Tom Parke, posting in rec.arts.dance, offered the following definitions: