In the phase immediately after the yeast is added to the sweet wort, it needs to multiply to ensure a good fermentation. To do this effectively it needs oxygen.
A lack of oxygen is not such a problem for kit or malt-extract brewers because they don't boil the full volume of wort, although aeration can still be beneficial. Tap water contains oxygen and the action of topping up the fermenter with water mixes oxygen from the atmosphere into the wort. So don't be afraid to fill the fermenter vigorously.
For all-grain brewers aeration is particularly important because the entire volume of wort is usually boiled and boiling drives off oxygen. There are several ways to ensure there's adequate oxygen in the wort for the yeast to multiply, before it turns its attention to making alcohol and carbon dioxide.
The simplest and cheapest way is to stir the wort vigorously with a long spoon or mash paddle for a few minutes, or until your arm feels like it's about to drop off. If you are using a plastic or glass carboy rather than a fermenter, you can cover the opening and shake it.
For mechanised aeration, some brewers attach a paint stirrer to an electric drill and give it a blast for a few minutes. This is an easy and effective method of aeration if you already have a drill. Paint stirrers to fit to a drill can be bought cheaply from hardware stores.
Many brewers use an air stone to aerate their wort. An air stone is a block of stainless steel with minute holes in it through which air from an aquarium pump — filtered through a HEPA filter to remove most micro-organisms such as wild yeast — or pure oxygen from an oxygen bottle is pumped. The holes are usually 2 micron, for use with an air pump or compressed oxygen, or 0.5 micron, for use only with compressed oxygen because aquarium pumps cannot produce enough pressure to effectively force air through a 0.5-micron air stone. If you have access to an oxygen bottle from an oxy-acetylene welding kit you can use that.