A unit of effective radiation dosage. It doesn't measure the amount of radiation you were exposed to or how much was absorbed (those are hard to distinguish, by the way), but the relative "effect" of the radiation dose. Some radiation is more harmful than others, and so on, and this must be taken into account. So you could have absorbed a lot of X-rays and suffer the same amount of biological effect as absorbing a lot less energy in alpha particles or something.
A "rem" stands for "roentgen equivalent man," and represents a unit of effect on human (or generally mammalian) tissue. The rem is the unit in the cgs system; the SI unit is the sievert. 1 Sv=100 rem.
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