There are two basic types of circulatory systems in animals that require blood be pumped through the organism to deliver oxygen and remove waste. There is an open circulatory system and a closed circulatory system.
In the most simple terms the closed system the blood typically does not leave the network of veins and fill up body cavities. The open system the blood is pumped into the system. It pools around organs and tissues and then is retrieved back into circulation. For example, think of backwash in a glass of juice. Take a sip, swish it around and let it slide back into the glass. That is essentially an open system.
While the open circulatory system is the least efficient of the two systems there is at least one advantage. Animals that have the open circulatory system are not as vulnerable to pressure. Mollusks can swim and dive to greater depths without the problems that arise with a closed circulatory system. Oxygen and carbon dioxide are served through separate systems.
The blood in an open system has to be a little different. It is sometimes called hemolymph. It is indeed a mixture of blood and lymph so that the cleansing and distributing of nutrients can be done while dumped around the tissues in the open circulatory system.
Another perceived advantage is that the animal has a great ability to control body temperature. Insects can survive in different environments because they are able to quickly get rid of the heat.
The most simple example of a closed circulatory system is in the earthworm. It is also the most inefficient of the closed systems. The earthworm has two blood vessels. They have a ventral vessel and a dorsal vessel. These carry the blood back and forth between the head and the tail. In the case of the earthworm the blood is not pumped but moved through the body through contractions called peristalsis. There are five pairs of vessels in the anterior of the worm that force the blood into the ventral vessel. Since the skin of the worm is so thin the exchanges of gases can be done through the skin as well.
Each type of animal as we go up the chain tend to have some different tools to make their closed system work more efficiently. Fish have gills and a two chambered heart.
Humans have a four chambered heart. Blood that is oxygenated is never mixed with blood that is de-oxygenated. The capillaries take care of the transfers of gases in the lungs and send the oxygenated blood back to the heart to be pumped out to the organs.