Hi Ina,
I am not a doctor, so please take this with a grain of salt, but I am a scientist who does imaging for living (in my case MRI), so I am a bit informed about the topic.

CT/CAT scans are basically fancy versions of X-ray imaging. CT stands for computed tomography. X-rays are used to make a three-dimensional anatomical image of the body area in question. Cancers can be detected by the swelling and anatomical distortion they cause, or by surrounding edema.

PET( positron emission tomography)is based on detecting pairs of gamma rays emitted indirectly by a radioactive biologically active contrast agent introduced into a body.
**( Technically, the contrast agent/tracer is unstable, decays by emitting a positron, which happens to be a particle that is anti-electron. When positron meets an electron in the tissue after traveling a little they annihilate emitting a pair of gamma rays, and an 3-D image can me made to show the origin of where those gamma-rays)**.
PET is used to detect areas of high metabolic activity in the body. It typically uses a molecule called FDG, an analogue of glucose. All the cells in our body use glucose for their metabolism, but some use more than normal. This can be perfectly normal, for example, our brain, which constantly uses a lot of energy. Also, cancers use more energy then normal cells, so this is how they are detected.
The radioactive dose one receives is very small so there is no need to worry. PET is pretty good at detecting metastases around the body, as well as primary cancers. It can not detect very small cancers.
PET suffers from poor resolution so sometimes when they do PET they will do a PET/CT, use CT images for anatomy and overlay PET ones on the top to show activity.

So, to summarize, CT will show if something looks bigger/has swollen etc (anatomy). PET will show if it has a high metabolic activity (function). The can be complimentary. Either of those can be indication of cancer, but doesn't have to be, things can be swollen/have higher activity and not be cancer. Biopsy is the only way to tell if something is cancer or not. Images tell us where to look.

In my case, I had both CT and PET/CT done. CT has detected my primary tumor but nothing in the lymph nodes in the neck as they were all the normal size/not swollen. PET, on the other hand, has in addition to the primary tumor, also detected activity in the lymph node area. After surgery, one of the removed nodes, pretty small one, 7mm, was positive for cancer.