Even a single puff of one cigarette can give you cancer. The more you smoke, the more your odds increase.
Tar, not nicotine, is what causes cancer. That's the stuff in the smoke. It's made up of over 4,000 different chemicals, 60 of which definitely cause cancer.
Cancers come in all shapes and sizes, but what it is is a group of cells which are multiplying like crazy, too fast for the rest of your body; thus tumors and other malformations are formed. When you smoke, you inhale all these carcinogens that mutate your cells. The more caked with smoke your lungs re, the more likely it is that your cells will mutate and form a cancer.