A linear effect means simply that two phenomena are related by a sloping line. Think of the relationship between gasoline consumption in an engine and distance travelled. That is a linear relationship and the line describing it shows a linear effect. More fuel more miles travelled.
Two issues bring the linear effect to my mind.
The first is the relationship between exposure to harmful radiation and detrimental medical consequences. There is not a linear effect. At low levels of radiation there does not appear to be any harmful medical consequences. People who live in Denver or fly professionally on commercial airlines see no medical consequences. Those people who survived Hiroshima and Nagasaki had longer than average lifespans.
On the other hand, people who smoke only a few cigarettes regularly and people who are exposed to moderate secondhand smoke do not have the consequences of lung cancer that are evidenced at higher exposures to cigarette smoke.
Neither of these are expected linear effects. We generally assume that something that is harmful in a large dose is less harmful in a smaller dose. That is a linear effects assumption.
It is often not true.