The word "rent" in this sense is not directly equivalent to its usual use meaning a payment on a lease, although it stems from Adam Smith's division of incomes into profit, wage, and rent.  Rent-seeking behavior is distinguished from profit-seeking behavior, in which entities seek to extract value by engaging in mutually beneficial transactions. 
Rent-seeking is often associated with lobbying for economic regulations such as tariffs. For instance, if FooCorp, a domestic producer of widgets, can lobby the legislature to levy a tariff upon widget imports, then FooCorp can sell its widgets at a higher price. If the legislature bans the import of widgets, or effectively bans them through high tariffs, then the additional price extracted can be quite significant. Collusion between firms and the government agencies tasked to regulate them can be a haven for rent-seeking behavior, especially when the government agency must rely on the firms for knowledge about the market.
The moral hazard of rent-seeking can be considerable. If a firm can calculate the cost of lobbying, bribing, or otherwise causing the government to enact a favorable regulation, then it can compare this cost with that needed to accomplish a similar benefit within the market -- for instance, by capital improvements or increased efficiency. If "buying" a favorable regulatory environment is cheaper than building more efficient production, then a firm may reap incomes entirely uncompensated.
Claims that a firm is rent-seeking therefore often accompany allegations of government corruption, or the undue influence of special interests.