Radiation
Radiation is a process in which energetic particles or energetic waves travel through a vacuum, or through matter-containing media that are not required for their propagation. Ionizing radiation (the only kind implemented in-game) is generally harmful to living organisms, as it mutates their DNA in random, unpredictable ways.
Sources of radiation
Radiation is found predominantly in craters, toxic waste dumps, and hazardous waste sarcophagus, but other, more unheard-of sources also exist; any tile that has contained radioactive gas or the small unmarked craters that dot the New England countryside are heavily irradiated.
In general, radiation is easily avoided. If for some reason you want to scavenge those irradiated locales, its better to keep trips short or to wear protective clothes. Wearing a clean suit or a helmetless suit of power armor will partially protect you; but if you want to loiter in the irradiated area for a long timespan, its recommended that you wear a Hazmat suit, ANBC suit, or a full set of Power Armor, as those items will confer complete protection from radioactive sources.
Detecting Radiation
Radiation cannot be directly detected by the survivor, but there are instruments with the ability to reveal the radiation level off both your body and of the ground in which you are stepping (the actual advantage gained from knowing the radiation level of a tile after you have been already contaminated by it is questionable). However, because the places in which radiation exists are well known in advance and far in between, obtaining them shouldn't be a high priority.
Radiation Sickness
Radiation sickness in game is represented by a myriad of penalties that start triggering once your body radiation level rises past the 80 rads threshold. At this point, you'll suffer a small penalty to movement speed, a one point reduction to the strength stat, and occasional vomiting; depending on worldgen options, you may also have a very small chance of gaining mutations. At 120 rads of exposure, you will begin to suffer penalties to every stat. Actual damage to all your body parts and random episodes of pain will begin once your exposure goes past 150 rads. Ultimately, an exposure higher than 500 rads is invariably fatal when left untreated, but with the liberal application of healing agents or with some luck (in the case you manage to mutate several times in a row), it can be survived.
Due to its nature, radiation sickness is very hard to treat. Your body will eventually recover naturally by slowly lowering its radiation level, and this rate can be increased by consuming iodine tablets prior to exposure but even then this is not very impressive. A specialized CBM exists which will decrease your radiation by 5 rads every time it is activated, but even that may not be enough to save you if you were exposed for a long time. And lastly there are Prussian blue tablets that help get rid of radiation, which are taken after exposure.
In short; getting irradiated is probably not a good idea. Unless...
Radiation for fun and profit
That having been said, there are reasons for merrily following the clicks of the geiger counter from time to time: the small cost of a couple iodine tablets can go a long way to kick-starting a young character.
For starters, Toxic Waste Dumps can be excellent places to obtain early game gems at relatively low personal risk- hard hats, tools, skill books, military badges and 30 gallon drums are just a few of the potential goodies that regularly spawn inside those maintenance buildings with only a locked wooden door, the usual wilderness threats, and ponds of mildly radioactive sludge standing guard. Just don't try to set up camp there, as appealing as that sturdy windowless bunker might seem: the highest levels of radiation can often be found inside the building.
If you should be lucky enough to stumble across a level 2 digging tool, craters can be similarly appealing places- the impact sites can often be mined for conveniently clustered rock and the occasional bit of metal. Just be sure to keep an eye on your exposure levels and take a break for a few days to cool off if you start getting above comfortable levels.
Lastly, radiation is the lazy man's mutagen: a fully healed character with a few first aid kits can easily survive a 200 rad exposure, and will be almost guaranteed at least one mutation along the way. Those feeling especially daring can also set up a nearby safezone like a car or small shelter just outside the irradiated area to recuperate between bouts of soaking up atomic sunshine. If one should be lucky enough to develop the Radioactive mutation, congratulations: you just hit the jackpot! Go home, and refrain from purifying that mutation until you've built up a couple of points of persistent radiation on a tile where you can sleep, study and craft for long periods of time. The next time you want to roll the genetic dice, all you have to do is hang out in your nuclear tanning bed for a while in relative comfort and safety (aside from the violent nausea and stabbing pains of radiation sickness). Keep a steady exposure between about 90 and 100 for a few days, and you'll be on the road to ascending to a higher form of life. Or to becoming a nightmarishly twisted zombie-annihilating crime against nature. Either one works.