The breathing system of spider is different from ours. There are two separate systems involved, book lungs and tracheae. The book lungs are assumed to be the first respiratory system evolved which is later replaced by the tracheae.
A book lung consists of a stack of plates. Blood flows through these plates and the gas is exchanged. The gases pass in and out by a diffusion process and it therefore not very efficient. In more active spiders tracheae replace part of the book lung or the whole book lung. The tracheae are tubes that do not branch but run from the opening on the outside directly into the tissues and organs.
*Copied by me of some page I found through google* before anyone thinks I'm a genious. :-) Although I am