The body is always making new blood vessels in a process called angiogenesis.  "Angio" means blood and "genesis" means beginning.  There are many normal reasons blood vessels are created, but cancer cells "trick" your body into providing blood vessels so it's tumors can grow. Here's the basic story:
How tumors grow
A malignant tumor starts from a single cell that has become cancerous.  This single cell divides and makes more cancer cells until there is a tumor.  These first cells can use the nearby blood vessels to get nourishment, but as the tumor grows, the cells in the middle of the tumor get farther away from the blood supply.  For the tumor to keep growing, it must have new blood vessels.  Without new blood vessels, a tumor can't grow larger than about the size of a pin head.
How cancer cells "trick" your body
Cancer cells release chemicals to prompt the growth of new blood vessels.  The cells that make up the lining of your blood vessels are called endothelial cells. The cancer cells prompt these endothelial cells to grow more blood vessels to supply the tumor with nourishment.  As more blood vessels are made, the tumor cells and the blood vessel cells feed each other's growth.  More blood vessels nourish the cancer cells as they multiply.  The cancer cells, in return, release more chemical messengers into the blood.  These chemicals cause the endothelial cells to grow and create even more blood vessels. So, the tumor gets larger and larger.
Anti-angiogenesis drugs
Anti-angiogenesis drugs don't attack cancer cells directly. Instead, they target the blood vessels the cancer cells need to survive and grow. By doing this, they may help prevent new tumors from growing. They may also make large tumors shrink if their blood supply is cut off.
Scientists have found a number of different pathways that cancer cells can use to cause blood vessel growth.  Each step in these pathways is a possible target for cancer treatment.  Different drugs are being tested to see if they work at different steps.  For instance, some stop the endothelial cells from receiving the chemical messenger from the cancer cells.
Combining targeted therapy with standard treatment
If your oncologist recommends anti-angiogenesis, he or she may also recommend chemotherapy. Combining anti-angiogenesis drugs with chemo or radiation seems to work better than using them alone. In some cases, anti-angiogenesis drugs shrink tumors, but in others they just seem to stop them from growing any larger, while chemotherapy often causes tumors to shrink or disappear. In some cases such as kidney cancer, anti-angiogenesis drugs may prove to be a better option than chemotherapy.
Because of the way anti-angiogenesis drugs work, they are only useful in treating cancers that form tumors. They won't work against blood cancers like leukemia.
Because this treatment is so new, it's not clear how long patients need to keep taking the anti-angiogenesis drugs.  Scientists are testing these newer approaches in ongoing clinical trials. 
Enzyme Inhibitors
Various enzymes in our bodies control nearly everything our cells do, even cancer cells.  Research has shown scientists which enzymes are giving cancer cells the "go ahead" to grow.
Drugs are being developed to inhibit the behavior of these enzymes and to block their signals, which keeps the cancer from growing and spreading. Then, chemotherapy steps in to kill off the cancer. 
Stopping cancers with out-of-control growth may also help people live longer.  Your oncologist may refer to these drugs as small-molecule drugs, signal-transduction inhibitors, or enzyme-inhibitors.
Apoptosis-Inducing Drugs
"Apoptosis" is the medical term for cell death.  Apoptosis-inducing drugs cause the cancer cell to die.
As a targeted therapy, these drugs are able to tell the difference between a healthy cell and a cancer cell.  Chemotherapy and radiation also cause the death of cancer cells, but they do so from the outside and don't discern between the healthy and the cancerous cell.  The targeting drugs are aimed directly at the parts inside the cancer cell that control the cell survival and death. They do this by changing the proteins inside the malignant cells.