Perhaps the easiest way to improve the quality of your homebrewed beer is to do as much as you can to ensure that your yeast is both healthy and happy. Happy yeast is happy beer - or at least a happy beer drinker.
Personally, one thing that I failed to account for early in my brewing career was the availability of oxygen as fermentation begins.
Yeast will ferment sugars into alcohol without oxygen. In fact, yeast will not produce alcohol during aerobic fermentation. Aerobic respiration (or fermentation) is the first phase of active fermentation, and occurs after the lag time associated with pitching yeast into a new environment.
During this time, the yeast scavenges available oxygen and ferments sugars - but the byproducts of this fermentation do not include alcohol (although CO2 is still produced in large amounts, so this phase can be detected by watching your airlock).
This phase of fermentation, which typically can be finished in as little as several hours, will produce many new yeast cells and significantly aid the yeast in its ability to continue to ferment the remaining sugars and produce our well-beloved byproduct of fermentation, ethanol.
So while your beer may be alcoholic without aerating your wort, (it will be - unless you have other problems altogether) it will not be as attenuated as it would have been with O2, and will almost certainly take longer to ferment than it would have with appropriate levels of oxygen. But what is the best way to dissolve oxygen into your wort, and when should this be done? Furthermore, how much is appropriate, and is there such a thing as too much?
First, and most importantly; brewers should only attempt to dissolve oxygen into their wort after it has been cooled to below 80°F. Methods for dissolving oxygen can vary. Simple methods such as agitating the wort and dumping cooled wort from (sanitized) vessel to (sanitized) vessel - i.e. boil kettle and bottling bucket - are the most common. Slightly more advanced aeration methods can involve aquarium pumps and pure oxygen cylinders used in conjunction with diffusion stones.
When using an aquarium pump, oxygen makes up only a small portion of the gas pumped into the wort - meaning that the pump should be run for longer (perhaps as long as an hour or more). When using pure O2 gas, ample levels of O2 gas can be achieved in as little as 5 minutes or less (assuming that diffusion stones are used, to increase the efficiency of dissolving the gas into solution).*
For those a bit more technically minded, 8 to 10 parts per million is a good level for dissolved oxygen in solution. However, the equipment for measuring these sorts of things are not common and most homebrewers (me) do not have access to these sorts of things. Fortunately, for our purposes, rough estimations typically suffice.
However, it is possible for the brewer to dissolve so much oxygen into his/her wort that levels are too high. These levels could be toxic to yeast or cause oxidation and/or flavor instability in the finished product. However, I have been assured by Neva Parker at White Labs that while this is a possibility, it is highly unlikely that the average homebrewer would be able to achieve this feat of beer ruination.
Moral of the story: 'Give your yeast what it needs, and it will give you what you need (or want)' Even simple and unsophisticated methods of dissolving oxygen into wort can yield an improved product along with shorter fermentation times and more attenuated beer.
*The numbers given for aeration times are for five gallon batches.