How is smoke harmful to the human body?

It really depends on the kind of smoke more than anything.
Cigarette smoke is a hazard due to things mixed into the tobacco, like tar and other chemicals. Because it is often inhaled directly, and pushed into the lungs, it can affect the cells even in the deepest portions of the lungs. Chemicals in cigarettes can be mutagens, and cause mutations in the cells of the lung, possibly causing cancer.
Smoke from house fires can be dangerous, as it occupies space in the lung, preventing oxygen from reaching the bloodstream. In addition, smoke can be heated to the point of damaging and burning cells, causing them to die.
Some smoke contains carbon monoxide, which can bond to red blood cells, and displace the oxygen that needs to reach the cells.
Some smokes, like CS gas, can contain chemicals that damage the lung itself. Mustard gas, as an example, can cause blisters to form in the lung. When these pop, the patient may drown. Gas weapons are extremely dangerous.