You're getting a lot of BAD ANSWERS!
First of all, smoking CAUSES cancer. To say that it doesn't cause cancer in all smokers, therefore it's not a cause, is a complete logic failure. Matches cause forest fires. However, not every time a lit match starts a small blaze in the woods does it lead to thousands of acres of damage. To call a match "a contributing factor" to forest fire would be completely stupid.
There are a number of ways that carcinogens can cause cancer. Radioactive elements, if present in cigarettes at all, are not the main factor. Tobacco plants don't MAKE radioactive material, and unless they happen to soak it up from the soil or some other source along the production line, that's not going to be an issue at all. If there's polonium in the soil in the carolinas, then all the farmers are screwed, not just the tobacco growers!
In order to turn cells from normal to cancerous, a series of changes have to take place in their genetic material (DNA). First of all, a series of mechanisms which regulate the rate at which cells grow and divide has to be defeated. Cells that start off as normally developing a nice, even, one-cell thick covering layer along the lining of the airways are then influenced to grow into heaps and lumps - "tumors" (tumor literally means bump or lump).
Just because it starts growing at an uncontrolled rate, though, doesn't make this new tissue into a cancer. Cancers are "invasive". This means that as the cells divide, the new daughter cells don't stick to the spot where they originated. They break across layers of adjacent tissue and start reproducing into layers of material in which they have no business. Cancers also "metastasize" which is to say that they can start cropping up in parts of the body not connected in any way with the location that spawned this new cell line. Growths of these cells can be found in the lymph nodes, liver, bone marrow, brain, and many other sites. The ability for the new cell type to invade into other tissues, and the ability for it to break off, float away in the current and start new growths are what define something as a malignancy - a cancer. For this to occur, a series of changes have to take place in the actual DNA that acts as the blueprint for cell production and cell life.
Carcinogens are chemicals or cellular-environmental factors which influence these types of genetic changes. There are a number of ways in which this can occur.
One method is to physically combine with the DNA through a chemical reaction. Direct alteration of the DNA would normally be detected by a complex series of DNA error-checking mechanisms in the cell. However, if the error checking mechanism itself is what gets damaged, this can be the first of a series of changes that lead to cancer! Radiation damage causes direct chemical changes to DNA also. It damages all kinds of molecules in the living system, but when DNA gets damaged, it's a bit more of a problem than if some protein in your cartilage gets cooked. Protein damage does occur, but if the system is healthy, it just gets replaced or repaired. Once DNA gets damaged, the instructions for "how to repair" are altered, and a good copy of these instructions may not be available.
Another way is for chemicals to "intercalate" into the DNA. Flat molecules that are about the size of DNA building blocks can sometimes get wedged in between the complex parts of this enormously large molecule. It's not unlike having a bit of your shirt get caught in the zipper of your jacket! If there is a bit of extraneous material jammed in the DNA, and it gets copied without that bit removed, then the whole sequence downstream from that point is screwed. Many times, this would just lead to a completely non functional cell as a result, but every now and then it could be the first step toward cancer.
DNA replication errors can occur spontaneously. The system can't be 100 percent perfect! However, its really really really REALLY CLOSE to perfect. Billions of cell reproductions go on in every person in a life time. Not everyone gets cancer! If you think about it, thats already amazing!! However, you can increase the likelihood of spontaneous error by increasing the rate and frequency of cell turnovers. The most common way to do that is to cause ongoing low level injury. Any part of the body that gets abnormally chronic abuse ends up adapting by higher rates of cell turnovers. In addition to other chemical routes of DNA damage, smoking causes an interesting phenomenon related to exactly this.
"Metaplasia" is a phenomenon where one tissue type turns into another in response to a chronic irritating stress. Smokers bombard their airways with hot microparticles of partially burnt leaf many times a day. The lining of the airways, which is normally a moist, mucus secreting barrier layer, is overwhelmed by this. It gets torn and damged constantly. It tries to heal these spots with high rate cell reproductions. Eventually it begins to produce a cell line which is more like the skin on your outside. It produces "squamous metaplasia". Squamous cell linings are not usually found in the airways, but they are present in chronic smokers - perhaps as an adaptation to the more abraisive treatment.
Interestingly, lung cancer is frequently of the squamous epithelial cell carcinoma type. This is interesting, because squamous cells are not typically found in the location where these cancers occur! The squamous cells are a result of metaplasia, the response by the body to chronic irritation.
The bottom line is that numerous chemicals in cigarette smoke can (and DO) directly combine with DNA, causing genetic alterations in the cells lining the lungs. Other chemicals exist which clog the structure of DNA and cause it to mis-replicate - leading to other kinds of genetic errors. The constant irritation to the lining of the airways causes increased rates of cell reproduction and more chances for genetic errors to occur. Eventually, if all the right (wrong) factors occur in the same cell, it turns into a cancer. This cancer is then directly caused by smoking, just like the forest fire is caused by the match.
But that's just the tip of the ice burg!
If you're curious, why not ask about smoking and vascular disease, or smoking and connective tissue degeneration? These deadly results of smoking are far more common and far less recognized!
As an aside - after a smoker quits smoking, the cancer risk doesn't drop down to the rate of the non-smoking population for about 20 years. The reason that happens is that the general rate of cancer goes UP as we get older, and the general non-smoking populations cancer risk INCREASES to meet up with the former-smokers slightly decreased risk.
If every smoker in the world quit smoking today, my career would still be populated in large part by taking care of smokers diseases till I neared retirement!!!
I hope that answers your question.
I'm a surgeon.