Ballet is a special kind of dance that was originally from France. Ballet has a lot of technical movements with their own names, like pirouette and battement. Ballet is usually danced to classical music.
Dancing is a way of moving that is usually set to music and is pleasant to watch. There are many different kinds of dance, from very formal dances like ballet that have defined movements, tribal and traditional dances from many cultures, through to unstructured dancing that young people do at nightclubs.
A dancer is a person who dances. Some people are employed as dancers, which makes them professionals. The woman in the picture is a ballet dancer.
Twerking is a provocative dance where a person squats or bends over and wiggles their hips, usually to hip hop or rap music.