Typically for most organs, an artery takes freshly oxygenated blood into an organ from the heart, while a vein takes the de-oxygenated blood away from an organ back to the heart.
The liver is an exception. The hepatic artery supplies oxygenated blood to the liver from the heart. The hepatic vein takes away deoxygenated blood from the liver back to the heart. However, the hepatic portal vein takes blood from digestive organs (intestines, colon, etc) and deliver the newly absorbed nutrients from food to the liver for further processing (metabolism). Therefore the liver is one of the few organs that has a 'dual-blood supply' simply because it receives blood from the heart (hepatic artery) and from the digestive organs (hepatic portal vein).
There is another vessel inside the liver that does not carry blood (bile duct). The liver produces a substance called bile, which is used to help break down and digest fatty foods. The bile duct carries bile secreted from the liver and delivers it to the gallbladder, which stores most of the bile in the human body until it's needed when we eat food that contains fat.