"Japanese form" is a flower with five or more outer petals and a center of staminodes (stamens without pollen) -- a single row of large petals and a center of colorful stamens.
[From Peonies, by Jane Fearnley-Whittingstall, p. 208:] In China today there are nine recognized flower shapes for peonies; they can be translated as single-petal, lotus, chrysanthemum, rose, sweet-scented osmanthus-holding, golden-ring, crown, silk-ball and pavilion... [four categories are] recognized by the Royal Hotricultural Society: single, semi-double, double and Japanese (also called imperial or anemone form).
[Ibid, p. 61:] The outer, guard petals are broad and rounded. At the centre the petals are ribbon or thread-like and are known as petaloids or staminoides.
[Ibid, p. 63:] They consist of one or two rows of large outer, or guard petals, surrounding a more or less showy central display of densely packed, narrow petaloids or staminoides. They are sometimes the same colour as the guard petals, sometimes a contrasting colour. When the petaloids almost completely fill the flower, it's form is sometimes described as 'anemone-flowered'.