Reusing yeast is certainly one of those things that makes sense from a cost-savings perspective. Indeed, brewers have been harvesting yeast crops at the end of fermentation, temporarily storing the yeast and re-pitching it into subsequent batches for a very, very long time.
One thing that has been demonstrated through research is that brewing yeast is not something found in nature; mutations over thousands of years of brewing have resulted in yeast strains that are only found in breweries. This is really not too different from domesticated animals.
Another thing that is known about brewing yeast is that some of the properties strains are selected for by brewers change with subsequent generations. One example is a yeast strain’s flocculation characteristic; lager yeasts, in particular, tend to become less flocculent if the yeast is used for too many generations. Some strains change faster than others, but as a general rule most breweries do not continue to reuse yeast indefinitely and new yeast is brought into the brewery by growing cells from some sort of storage form (liquid nitrogen, freeze dried samples or cultures stored on a growth medium).
The best yeast to reuse comes from a healthy and normal fermentation. Batches that do not ferment normally, or those that have unusual aromas, are not top candidates for cropping. Assuming that you have a batch that is a good candidate for yeast cropping, you need to determine how to harvest the yeast. Most yeast strains, including most ale yeast, will eventually settle to the bottom of the fermenter within a few days following the end of fermentation. I think the easiest way to harvest yeast at home is to rack the beer out of the primary, swirl the yeast up from the bottom of the fermenter and pour it into a clean and sanitized storage container. Most thick slurries harvested in this manner contain about 750 million cells per milliliter.
It is best to store yeast in a vented container because there is a real possibility that the cropped culture will produce carbon dioxide. Do not store yeast in sealed glass containers as this may result in exploding glass containers. I am a proponent of using plugs of cotton batting to close the mouths of Erlenmeyer and Fernbach flasks.
Yeast cells have a finite storage life, and as energy reserves, such as glycogen, are consumed during storage, cells begin to die. The most common method used to extend the storage life of yeast cultures is to rapidly cool the yeast culture and store it somewhere around 32–38 °F (0–3 °C). Some brewers wash yeast with cold water after harvesting to dilute the beer content of the slurry since the alcohol content of beer is detrimental to cells during storage. Yeast can easily be stored for up to a week in this temperature range without losing too much viability in the culture. Anything greater than a week is too long for most commercial brewers because the economic risk of using old and tired yeast cultures is simply too great.
Things are not too different with homebrewing, except the risk of failure. If I am running a brewery that brews 100-barrel batches and ferments these batches in 400-barrel fermenters, the cost of a failed batch is equal to about 20 man hours of work plus about 20,000 pounds of malt and 200 pounds of hops; roughly $8,300 in labor and raw material costs.
At home the cost of failure is perhaps even greater. You spend your hard-earned free time and you pour your heart and soul into crafting that perfect batch of wort. If you want to re-use yeast at home, do not push the limits of storage time and expect anything miraculous to occur. In fact, you should actually expect poor results because that is what you could see.