Real Ale

Cask-conditioned beer is brewed from only traditional ingredients and allowed to mature naturally.
The unfiltered, unpasteurised beer still contains live yeast, which continues conditioning the beer in the cask (known as 'secondary fermentation'); this process creates a gentle, natural CO2 carbonation and allows malt and hop flavours to develop, resulting in a richer tasting drink with more character than standard keg beers.
Real ale is always served without any extraneous gas, usually by manually pulling it up from the cellar with a handpump. This is the traditional way of brewing and serving beer; only a few decades ago did filtered, pasteurised, chilled beer served by gas become normal. The only place in the world where cask-conditioned beer is still commonly available is Britain.

Keg beers are generally sterile filtered and pasteurised as part of the brewing process. This kills the yeast, preventing any further conditioning, and the beer is then racked into sealed, gas-pressurised kegs. Such beers generally taste blander than their cask-conditioned counterparts, and the use of flash-chillers or cold rooms (*very* cold!) is standard as part of the serving process. That said, some microbrewers rack cask beer into kegs - though these are usually served with extraneous gas.
Keg beers have a much longer shelf life, especially when compared to a partially full cask. Real ale has to be manually vented and tapped, and left to settle (or the customer gets a cloudy pint due to the presence of yeast and protein - though harmless if drunk like this). Also, real ale will start to taste of vinegar (known as 'oxidising') if left in a part-full cask for too long. This is caused by acetic acid forming from a reaction with oxygen in the atmosphere.

In Britain in the 1960s and 1970s, a mass attempt by that country's major breweries to replace cask-conditioned beer in their pubs with keg was halted by widespread public outcry - largely credited to the rapidly-growing Campaign for Real Ale ('CAMRA'). Today in the U.K., cask-conditioned beer can still be found in most pubs thanks to the campaigning efforts of the drinking public. However, complacency would lead to the profit-hungry major brewers quietly and gradually phasing it out - hence CAMRA currently has about 70,000 members. In Ireland, the popularity of Guinness and lack of competition has meant that cask-conditioned beer is less common than in the U.K; however, availability there is now slowly on the increase.