IV contrast increases the radiodensity of structures with a high amount of blood flow. In other words, because IV contrast goes into your blood vessels, tissues that either ARE blood vessels or are highly invested with them will be brighter than they are without contrast and also brighter than surrounding, relatively non-vascular tissue.
Since the technology of CTs and x-rays depends on visualizing the body by creating interfaces between structures of different densities, contrast CTs can both create interfaces and obliterate them. Thus, your indications and contraindications to CT with contrast studies depend on what tissues you are imaging and what you are trying to find.
In pancreatitis, for example, the normally vascular pancreas will "light up" on CT with contrast, but dead areas will not. This allows you to estimate degree of necrosis involved in an acute pancreatitis.
You can also image intraabdominal bleeds because the IV contrast will spill out of lacerations into the abdominal cavity, you can use the interfaces between dense and lucent created by contrast to image hyper- or hypovascular lesions (such as malignancies, which can be either, or abscesses, which are lucent), or you can use contrast studies to search for intravascular filling defects such as when searching for pulmonary embolism or trying to image an aortic dissection.
Hope that helps.