There doesn't appear to be a good Wikipedia entry for RTB for me to link at the moment, when I want to blog about it so I'll draft my own explanation here. (Edit: there is an entry now, but I like my characterization better!) Keep in mind while reading this that I'm looking at RTB as a software engineer with an interest in economics, rather than as an ad industry veteran!
Real-Time Bidding (RTB) is a way for advertisers to pay publishers to show ad impressions on their websites. In this context, an 'advertiser' can be anyone with a product to sell: a car company, a small e-commerce website etc. A 'publisher' can be anyone with a website: a newspaper, a blogger, Facebook, Google etc. An 'impression' is a single instance of a banner ad being shown on a web page. The term RTB is usually not applied to other ad formats such as text ads ads, or ads alongside Google search results or Facebook news feeds, as different technology is used to buy and sell those other formats. The word 'bidding' comes from the fact that the price of impressions is set via an auction mechanism, where the role of the auctioneer is played by an 'ad exchange' who mediates between advertisers and publishers. The term 'real-time' stems from the fact that advertisers bidding in the auction must specify the price they are willing to pay for each impression separately for each impression, and within a strict time-constraint: usually around one tenth of a second, so the auction can run as the web page on which the impression will be shown is loading.
In terms of the technical aspects of participating in the auctions as a bidder, a computer subscribes to a stream of many thousands of bid-requests per second from the exchange, each one representing an impression to be shown very shortly. For each one, within the hard-real-time constraints, the computer must respond by specifying if/how much it wants to bid, in units of millionth of a dollar (u$). The cost of online advertising is usually discussed in terms of cost-per-mille (CPM) in dollars, so $1 CPM implies 1 million microdollars for 1 thousand impressions, or u$1000 per impression. The bidding computer can make its decisions based on the various attributes contained within the bid-request: the URL, the position of the ad spot in the page, the user's location and local time, the number of times this user has previously been shown a given ad, any additional data the bidder has on file or has purchased on that particular user, etc.
From my reading on auction theory, here is how I would characterize what goes on in most RTB auctions:
These characteristics make the RTB marketplace fairly different from, say, a stock exchange, where people buy and sell large pools of indistinguishable units of stock, whose value is intimately linked with the price at which it can later be resold, and in general no one cares who is buying or selling.
Participating in RTB auctions is fairly technically challenging, and good bidding strategies are not obvious, so there must be some perceived advantage over the alternative. In this case the alternative is a complicated web of relationships between publishers and advertisers, sometimes with 'ad networks' and other middlemen in between. The purpose of RTB is to set up a more efficient marketplace for advertising inventory. One where fine-grained control over bids allows advertisers to save money by purchasing a smaller quantity of better targetted ads, even if they pay more per impression. One where publishers are able to earn more from their web properties because they can more economically reach the advertisers who are willing to pay a high price per impression for the ads they do show. And one where the end-users who actually see the ads are happier because they see fewer cheap and irrelevant ads. Now obviously, for buyers to spend less and sellers to earn more something has to give, and that's where the word 'efficient' comes in. Right now there is a lot of opportunity cost and operations cost on both sides of the market and RTB is meant to reduce a lot of that. Whether or not any of the promise of RTB is actually materializing for any side of the market equation is still an open question, and it's a rapidly-evolving area of the online advertising industry.