A paper barrel is an oil cargo which is sold and traded on the open market, but not actually shipped; essentially, the cargo is passed back and forth on paper, hence the name "paper barrel." Obviously at some point the cargo of oil will be delivered, but it may have changed hands several times in paper barrel form before delivery actually happens. Trading in paper barrels, as you might imagine, can drive the price of oil up considerably, and in periods when oil prices appear to be rising precipitously, many people suspect that trading in paper barrels may be the culprit.
The idea of trading goods on paper but not actually taking delivery is actually fairly old, as is speculation in such cargoes. Many areas have a long history of futures trading for things like pork bellies, wheat, corn, and so forth. Futures trading had its origins in farmers who would travel to an urban area to get contracts to sell their crops in advance of actually having a crop to sell. By reaching an agreed-upon price early, many farmers hoped to ensure a base profit for their crop, but they were also making a bet that crop prices wouldn't rise substantially.
In turn, the person who held the contract could decide to sell it. Over time, futures trading began to dominate many markets, and today many major goods are traded in the form of futures, including oil. A paper barrel is simply a contract for oil futures, bought, sold, and traded in the hopes of making a profit.
Like other trades in futures, oil futures rely heavily upon speculation. Prices may go up in response to natural disasters, refinery problems, and projected low output, and they can go down when a surplus in predicted. Everyone who holds paper barrels hopes to profit from them, and few have intent of taking a “wet delivery,” an actual delivery of oil. Since most trading in paper barrels occurs in investment firms, banks, and stock exchanges, most people who trade in oil futures actually have only abstract ideas about the processing, refining, and shipment of oil, and they would have no idea of what to do with an actual cargo of oil.
Each time a paper barrel is sold, the value usually goes up, unless someone takes a loss in the interest of avoiding further losses. When trading is hot, it can drive oil prices up quickly, causing per barrel prices to jump radically, and in turn triggering a rise in fuel prices. Some people have suggested that speculation in oil futures should be curtailed, in the interest of avoiding price inflation, but others argue that it is simply a function of the free market, and that the market will eventually correct itself if prices become dangerously high.