The TL;DR answer is: yes, radiation can cause cancer, but no one knows for sure exactly what the risk is of one CT scan.
Radiation (especially in fetuses/children) increases the likelihood of cancer. Our information comes mostly from atomic bomb survivors, people exposed at Chernobyl, people treated with high doses of radiation for cancer and other conditions, and people exposed to high levels of background radiation, e.g. uranium miners.
Can one CT scan increase your chance of developing cancer? Yes, but how significant that is remains debatable. It depends on a lot of unmeasurable things: genetics, age, one's ability to repair the damage, which area of the body is being dosed, whether there are other carcinogens at work (e.g. viruses or co-carcinogens), etc.
The following are commonly found numbers. Given 10,000 people, about 2000 will die of non-radiation related cancers. If you expose those 10,000 people to 10 mSv of radiation, you can expect 5-6 additional cancer deaths in that 10,000 (or .5% increase).  However, if you expose someone to 10 mSv twice, with two months in between, the risk does not double.
The average dose of radiation per year from background sources is 1.5-3.5 mSv/yr.  Because of our ability to repair damaged DNA over time, it's generally estimated that, from natural background radiation, the risk of developing cancer from background radiation is about 1% of our total cancer risk.
The average CT scan is going to give you 7 mSv of exposure (that's a high estimate. The newer the machine, the lower the dose - usually.) What is your chance of getting a lethal cancer from that? The answer is: no one knows.
Relative risk has be calculated by some. The following all give you a (estimated) 1 in a million chance of dying from that event:
- Smoking 1.4 cigarettes (lung cancer)
- Eating 40 tablespoons of peanut butter
- Spending 2 days in New York City (air pollution)
- Driving 40 miles in a car (accident)
- Flying 2500 miles in a jet (accident)
- Canoeing for 6 minutes
- Receiving 10 mrem (.1 mSv) of radiation (cancer)
So, you can estimate that (maybe) the risk of dying from a 7 mSv CT scan is about equivalent to driving 2800 miles.
The point of all this is, no one knows your exact risk of developing a lethal or relatively non-lethal cancer (yes, that does exist) from an imaging study. Most ethical radiologists will tell you there is no known safe radiation exposure, and that one should avoid unnecessary radiation exposure. While some imaging studies are unnecessary, some of them are not, and the risk of avoiding the study greatly outweighs the risk of undergoing it. What you can do to protect yourself:
- ask if an MRI will give the doctor the same information (no radiation, but 2-3x the cost, longer wait)
- ask the doctor if the test is absolutely necessary (a lot of tests are ordered to cover one's ass)
- ask if there is another way to obtain the same information (e.g. colonoscopy instead of barium study)
- get your study from the newest scanners available (radiation can be reduced up to 60% between generations of CT scanners)
- reduce conflict of interest: any CT scan ordered in house (or by a doctor that owns a share in the imaging facility) should be questioned (but not necessarily refused.)
 USEPA Radiation and Health
 Radiation and Life
 Higher accumulated doses of radiation might produce a cancer which would only be observed several - up to twenty - years after the radiation exposure. This delay makes it impossible to say with any certainty which of many possible agents were the cause of a particular cancer. In western countries, about a quarter of people die from cancers, with smoking, dietary factors, genetic factors and strong sunlight being among the main causes. Radiation is a weak carcinogen, but undue exposure could certainly increase health risks.
 American Nuclear Society
 Risks of Radiation
 Radiation and Risk
From previously given answer on Biology.SE