How do I take apart a wooden pallet without cracking the wood?
They make several styles of tools that are made specifically for getting nails out. One tool called a "cats paw" looks like the picture below. (You may find similar tool under other localized names). A hammer is used to pound the hooked end in under the nail head so that it can then be levered out. 



Another type of nail puller is also very effective in that the tool is used by itself without a need for a separate hammer. The handle on this type of nail puller is used to slide up and pound dowwards to embed the jaws around the nail head. Pulling sideways in the direction of the "foot" forces the jaws to clamp tightly on the nail and pull it out.