The term hardwood is one of the more confusing terms in woodworking. It does not actually refer to the hardness of a wood. Scientifically speaking, a hardwood is an angiosperm. This basically means that is has covered seeds. Softwoods on the other hand are gymnosperms.
If you want to make the classification easy, just think conifers as softwoods and other trees are hardwoods. A few popular examples of hardwoods are oak, cherry, maple and birch. A few popular softwoods are pine, spruce and cedar.
If you were to look at the hardness of all of the hardwoods and softwoods, you would notice that some hardwoods are softer than some softwoods. For example, southern yellow pine is often used in decking as well as some indoor furniture. It is a relative hard softwood. Balsa on the other hand is technically a hardwood even though it is the lightest and softest of all of the woods. Basswood and butternut are two more common woodworking woods that are actually softer than many softwoods.
More info on wood hardness