Context: insurance, risk theory, marketing
Premiums are moneys paid by one party (the insured) to another (the insurer) as part of a contract whereby the insurer will compensate the insured in the event of loss generated by a particular risk. Thus if the risk is death, then the premiums will be paid for a life insurance policy.
What is most interesting is how the premiums are determined.
Mathematically, a premium principle is a function that transforms a random variable R (the loss) into a deterministic positively-valued number Π(R). This function has to satisfy certain properties, such as the premium for a deterministic loss (i.e. R is of a degenerate distribution), is the loss itself, and that premiums are subadditive.
There are several functions that can be used as premium principles. Theoretically the simplest is the expected value principle, where Π(R) = (1 + λ) E(R), with λ being a risk loading to prevent ruin in cases of catastrophic claims.
Pricing of insurance products are driven by marketing pressures. Nobody is going to buy your policy if it is more expensive than others. So usually the marketing department will determine the most appropriate premiums to charge, and it is left to the risk managers (usually the actuaries) to figure out a way that would ensure that the company does not ruin.