The Gemara in Masechet Aboda Zara (43) discusses the prohibition of making and possessing certain structures and images. Thus, for example, the Gemara (both there in Aboda Zara and in Masechet Rosh Hashanah 23) writes that one may not build a structure modeled after the Bet Hamikdash, or any of the furnishings of the Bet Hamikdash, that has the same dimensions as the Bet Hamikdash or the corresponding furnishing. This means that one may not make a seven-branched, metal candelabra resembling the Menora that was used in the Bet Hamikdash. As mentioned, however, this applies only if the dimensions are the same as those used in the Mikdash. It is therefore permissible to construct small models of the Bet Hamikdash, even if they are built precisely to scale and accurately resemble the original Temple, since they are not built according to the same dimensions.
Many synagogues have a seven-branched Menora with electric lights. Hacham Ovadia Yosef (listen to audio recording for precise citation) rules that such structures are permissible, since there are no lamps in which one can place oil and wicks, as there were in the Menora in the Bet Hamikdash.
The Shulhan Aruch (Yore Dea 141:4) rules that it is forbidden to make an image of the moon, sun or stars. This applies both to protruding images as well as flat drawings. In fact, a watch bearing an image of the moon was once shown to Hacham Ovadia Yosef, and he ruled that for Sepharadim it is forbidden to own such a watch. It makes no difference whether the image is of a full moon or a half-moon.
Hacham Ovadia also ruled that it is forbidden to make or possess a full-body, three-dimensional image of a person. It is permissible, however, to draw a flat image of a human being, or to make a bust, which features only a head and neck. He adds that it is certainly permissible to take photographs, be photographed, keep albums of pictures in ones home, and hang up pictures, such as pictures of Rabbis which many people hang in the Sukka on Sukkot. Even pictures of complete human figures are permissible; only three-dimensional figures are forbidden if they are full-body images.
Hacham Ovadia notes that it would be forbidden to make or own a manikin, which is a three-dimensional representation of a complete human figure. In his work Halichot Olam (p. 285; listen to audio recording for precise citation), Hacham Ovadia writes that women who wear wigs may not have manikins on which to place their wigs. It should be noted, however, that generally women do not use full-body manikins for this purpose, but rather heads, which, as mentioned, are permissible.
In one of his responsa, Hacham Ovadia rules that despite the general prohibition against full-body representations of human beings, it is permissible to make, sell and own dolls used for play by children. He presents several arguments for allowing dolls, including the fact that they are not made for decorative purposes, but rather for children to play with. The prohibition against images of people relates to the possibility that the image may be looked upon as a kind of Aboda Zara (object of idolatrous worship), which of course does not apply in the case of a doll.
Summary: It is forbidden to make or own a precise replica of the Bet Hamikdash or one of its furnishings, but small-size models and imprecise replicas such as electric seven-branch Menorahs are permissible. It is forbidden to make or own a picture of the sun, stars or moon. One may make or own flat images of people, as well as three-dimensional structures of people that do not feature an entire body. Thus, one may own a bust or a "head" used for storing wigs. It is forbidden, however, to own or make a full-body, three-dimensional representation of a human being, such as a manikin, though toy dolls are permissible.
