This video from Veritasium is surprisingly interesting, involving visits to a number of the most high-radiation places on Earth, including Chernobyl, Fukushima, and an airliner in flight. It also builds up to a shocker of an ending, so if you have time to watch part of it, start about 3/4 of the way through, which will be in a hospital near Chernobyl.
If you do skip ahead you might be mystified by the banana graphics. Bananas are actually slightly radioactive owning to the fact that they contain potassium, which has a naturally occurring radioactive isotope. Eating a banana exposes you to about 1/10 of a microsievert of ionizing radiation. The graphics represent how many microsieverts you would be exposed to in an hour in the various locations shown. For example, 20 bananas means an exposure of two microsieverts per hour.
Something the video doesn't get into but is worth noting: "Ionizing" radiation is any sort of radiation that can knock electrons loose from atoms and molecules, and hence directly create chemical changes, including damage to DNA. Not all radiation is ionizing, and not all ionizing radiation is equally dangerous and the danger is also very dependent on the type of exposure.
For example, alpha particles, which are emitted by some types of radioactive decay, are the bare nuclei of helium atoms. Outside the body they aren't very dangerous because they're very easily blocked. Generally even a sheet of paper or your outer layer of skin (made up mainly of dead cells) can stop a fair amount of alpha radiation, and a Geiger counter often can't register alpha particles unless its detector is equipped with a window able to transmit rather than blocking them. On the other hand, if you inhale a bit of radioactive dust that emits alpha particles, this can lead to serious harm to the unshielded cells of your lungs.
Photons are the particles of electromagnetic radiation, something that includes visible light, infrared, ultraviolet, radio, x-rays, and gamma rays. The energy of a photon depends on its wavelength, and X-rays and gamma rays have very short wavelengths and hence very high energy. X-rays and gamma rays are ionizing radiation and can definitely damage DNA and thereby cause cancer, though the danger of course depends on exposure. Ultraviolet light, while not ionizing, can still cause chemical changes that can lead to skin cancer, which is why you should wear sunblock outside and definitely avoid tanning beds. Ultraviolet light can also damage your eyes. Bright visible light, especially near the short-wavelength blue part of the spectrum, can also damage your eyes if you're exposed to enough of it, of course.
But the longer the wavelength, the lower the possibility of health effects. All wavelengths of light can cause heating, which can cook living tissue if it gets too hot, but aside from that, for wavelengths longer than visible light there's no known cause for concern for anything other than heating. Every object around you actually gives off a glow that depends on its temperature, but we don't see it because they glow in infrared wavelengths, longer than visible light. This is why we tend to associate infrared light with heat, but otherwise, contrary to popular belief, there's no special connection between infrared and heat. Any wavelength can heat things up, including microwaves, infrared, visible light, ultraviolet, and even X-rays. Even longer and less energetic than infrared light are the various wavelengths of radio.
About half of the older airport whole-body scanners in use in the United States (the sort where you stood between what looked like a couple of monoliths) employed low-energy backscatter x-rays and hence presented a small but non-zero risk of skin cancer (a lot less than the flight itself, however). But modern scanners use millimeter waves, and there's no evidence they're harmful.
(The text below the video has been revised a few times since the original post to correct typos and add clarity.)