Blood flow has 2 main functions: It carries the nutrients necessary for your cells (oxygen included) and it removes the waste produced by your cells (including CO2). When blood flow stops, your cells start dying.
As your heart pumps out blood, it has to get blood to every cell in your body, but it prioritizes everything. Before sending blood to your fingers, for example, it prefers to send it to more important organs. The 3 organs that get "dibs" on the blood are the brain, the lungs, and the heart itself. (Although the heart is pumping out blood, it needs blood too so that it can function.) If the artery that sends blood back to the heart gets clogged, it's called a heart attack. If the artery that sends blood to the brain gets clogged, it's called a stroke. If the artery that sends blood to the lungs it's called a pulmonary embolism (much less known than the other 3, but still very dangerous). If any of the organs stop getting blood, they will start dying. This is especially dangerous if part of the heart dies since that will prevent the heart from sending out blood efficiently in the future.
As you start bleeding out, your heart will start rationing where it's sending the blood. If you lose a lot of blood, the heart will still send blood to the heart, brain, and lungs, but it won't necessarily send it to, say the stomach or your arms. People will feel like throwing up (if the stomach loses blood) and will lose feeling in their extremities.