This is a poorly named but important concept in microeconomics. The idea was originated by Gordon Tullock in 1967 and the name was applied by Anne Krueger in 1974.
Rent seeking is the attempt to obtain personal benefits through governmental policies. It is a political phenomenon and an example would be a licensed doctor lobbying to make stricter the requirements to become a doctor. This limits the doctor’s competition and accrues more value to them for the same services and no increase in productivity.
Rent is a term that has long been around in economics (attributed to Adam Smith by some and David Ricardo by others). It refers to the payment to a factor of production in excess of what is sufficient to keep the factor in use. By this definition I would be “rent seeking” if I asked for a salary raise from £60k to £70k even if I would still stay and work as hard on the £60k. In this sense there is nothing wrong with “rent seeking”.
The term is actually used differently. It encapsulates the behaviour of bigger companies lobbying governments to enforce policies which increase their monopoly in an industry. It also describes the political actions of people who look to benefit from increased social welfare. David R. Henderson has proposed the use of “privilege seeking” as it better elucidates what is meant.