Market power is when a firm can charge more than the cost of making a new unit, because of barriers to entry. In a competitive market where firms rent their capital, they can only charge marginal cost: if not, other firms would pop up and steal the market by undercutting them. Some of the most competitive real live markets are close to this—supermarkets make tiny margins and offer near-identical prices. But most markets are to some degree away from this.
This means there are rents. In an extreme case these are monopoly rents. Imagine you own and run a railway between Manchester and Leeds. Building and operating a new one costs a lot, and this new railway would have to charge lower prices to attract customers. Assuming there are no other modes of transport then this gap—expected potential profits—determines what prices the monopolist can set before they'll face competition. The gap between a competitive price and the monopoly price is a rent. This is a pure redistribution from users of the network from whoever happens to own it.
It helps to distinguish between barriers to entry and costs of entry. If building a new railway was trivial technically, and the only cost was regulatory—e.g. you had to pay off a corrupt bureaucrat but rails appeared magically—then potential competitors face barriers to entry. Real costs that society has to bear—like using workers, capital, and management or entrepreneurship to organise building a railway line instead of doing something else—are costs of entry.
Barriers to entry reduce competition with no corresponding benefit, but when there are large costs of entry, the loss of competition is balanced by keeping resources spare. Competition requires "unnnecessary duplication", something that greatly troubled early socialists and led them to believe that the socialist economy would not only be more moral than the capitalist one, but more efficient.
In any case, a canny monopolist will set prices such that no competitor enters, enjoying their rents for as long as possible. Modern regulators aim to set prices so that natural monopolies do not earn rents, and consumers get higher consumer surplus instead. So far, so sensible. But this approach may have a key defect: it requires Herculean feats of innovative winner-picking elsewhere when you think the economy might be dynamic, not static.
This is because rents that are caused by costs of entry (but not those caused by barriers to entry) are automatic prizes that reward entrepreneurs in direct proportion to how much they can alleviate inefficiency in the marketplace. Higher rents not only motivate investment, as in the example above, where someone might build a second railway (or indeed a road, canal, bus service, or air link) between Manchester and Leeds, but also innovation.
This is quite general. Some innovation does not take away profits, but instead reduces the need for labour, land, or capital. But the cost of these is a rent too—a factor rent. Just as we don't see these as fixed, nor should we see market rents as fixed. As long as we think innovation is reasonably possible, we should be open to allowing market rents to exist to direct innovation. Efficiency-enhancing innovations do not fall from heaven—they come from where we focus our research activities as a society.
It's obvious how this applies to intellectual property. When I have a patent to produce a drug, if this drug is useful I will earn large rents. But it is precisely those rents that indicate that finding a substitute is so valuable. If a firm can find a close substitute, they stand to get some of those rents for themselves—driving innovation into the highest value areas. So we should be more sanguine about less competitive markets and natural monopolies—where competition is restrained by facts of the world, not regulation—they help tell us where improvement is most valuable. And they pay people for doing that improvement!