Yeast pitching is the process of inoculating the freshly-brewed wort with an appropriate, measured amount of yeast. The proper amount of yeast to pitch varies from style to style, from recipe to recipe.
The standard rule of thumb is one million active yeast cells per milliliter of wort per degree Plato. Some authors suggest more, others less. Moreover, authorities differ as to the appropriate amount for ales vs. lager beers; George Fix, for example, specifies 1.5 million cells per mL per ºP for lagers and 0.75 million for ales. Ray Daniels sticks with the in-between of the rule of thumb, which works well in virtually all circumstances.
 A Practical Example
According to the rule of thumb, a wort of 12ºP requires 12 million active cells per mL. A five gallon batch contains about 19L (19,000mL). So, 19,000 times 12,000,000 equals 228,000,000,000 - that's 228 billion cells for proper pitching.
For reference, 1ºP is roughly equivalent to 1.004 specific gravity. In the example, conversion works to approximately 1.075; however, higher gravities after which it goes so wonky that it's better to consult a conversion chart. For most purposes, however, multiplying ºP by four and putting 1.0 in front of it will be close enough as makes no practical difference.
 Commercial Yeast
The commercially available liquid cultures contain on average 100 billion cells per large foil-pack or vial, according to hemocytometer tests conducted by brewing-industry professionals. These numbers were confirmed by the author's own study. According to the rule of thumb, that's an insufficient quantity to properly inoculate 5 gallons of even mid-strength wort; in fact, it's less than half.
According to Jamil Zainasheff, dry yeast has an average cell density of 20 billion cells per gram. By the rule of thumb, about 12 grams of dry yeast is needed to properly inoculate a wort of 12ºP wort. Conveniently, many dry yeasts come in 11-gram packets- almost perfect for a mid-gravity beer.
Some brewers use fresh, new yeast each time they brew. The wise brewer will require a starter. (See Yeast Starters for an in-depth lesson on making a starter.)
Others use a slurry of yeast from a previous ferment. Calculating how much slurry equals a proper pitch is easy. Approximately 25% of a slurry the consistency of pancake syrup is yeast solids. There are approximately 4.5 billion cells in 1 ml of yeast solids. Thus, 1 mL of slurry contains about 1 billion cells. So, to inoculate a wort of 12ºPlato, you need 228 ml of yeast slurry. Note: "Slurry" means slurry, not washed yeast! (See Washing yeast.) Nor does it mean simply knocking out fresh wort onto a yeast cake.
In order to ensure a strong start to fermentation, short lag time, and proper ester formation, be sure to pitch the right quantity of yeast.
 Further Reading
An Analysis of Brewing Techniques by George Fix
Designing Great Beers by Ray Daniels