Venetian chain, box chain: in most cases, they're the same thing. This chain is sturdy and low-key, suitable for men and women. However, it's best for wearing alone or with light pendants, because heavy pendants may stretch or twist the links. Box chain links are typically all the same size, and have sharp or rounded 90° bends, rather than oval or circular curves.
The links of box chain look like little squares with open sides, giving a shape that almost looks like a paper chain we made in grade school. It's been suggested that it is sometimes called Venetian chain because it originated in Venice, but I didn't find any reliable sources stating so.
While most people use the terms "box chain" and "Venetian chain" interchangeably, some people who use the terms for slightly different styles. To the people who use the terms specifically, you can't really see through the links in box chain. On a Venetian chain, the links are shaped such that you can see space through the links.
However, on both types of chain, you may see drawn chain, meaning the links are longer and rectangle-shaped, rather than square, as if they have been drawn through a drawplate. Here's an example of drawn box chain.
Book chain is related to box chain, except that instead of square shapes, the links are stretched into rectangles length-wise, which looks like the spine of a book. That's why the Victorians called it book chain. Book chain was popular in Victorian and Edwardian times, and the most extravagant book chains were engraved or bejeweled on the front side of each flat link. Here's a picture of book chain. I have also seen modern chains sold as book chains, that are simply rectangular components joined with jump rings - however, unless the links are folded at 90° like the traditional box chain, they don't fit the traditional definition of book chain.