Aeration is the addition of oxygen to the must. Degassing is the removal of CO2. These are techniques which are optional but, when used correctly, can significantly reduce the "hot" flavors produced during fermentation. This results in faster drinkable mead - decreased aging time to reach similar results.
Yeast need oxygen in the early stages of fermentation - during the lag and growth phase they will use it to reproduce. For this reason, it's beneficial to add oxygen before pitch, and regularly, roughly until 1/3 of the sugar has been metabolized. After this point, additional oxygen may lead to unwanted oxidation.
The easiest way to add oxygen is to mix air into the solution by stirring it vigorously. A few minutes of stirring by hand will suffice. If using a drill and mix-stir attachment, this will only take a few seconds. Continue reading, however, because rapidly agitating the must will also cause the release of CO2 - degassing.
There are a few reasons to degas your mead during fermentation. If you are staggering nutrient additions across the first few days, it is necessary to degas before you make an addition. The nutrient particles provide nucleation points to the dissolved CO2. This works similarly to Diet Coke and Mentos - the result is a mess. Degassing removes this CO2, which in high concentrations will stress your yeast during fermentation.
Initially, you degas your mead as you aerate it - stir the solution vigorously by hand or with a drill and mix-stir attachment. Be wary; it's easy to cause too much CO2 to come out of solution at once. In a carboy this can result in a geyser that hits the ceiling. In a bucket it will simply overflow.
When aging wine, you can speed up degassing with a vacuum degasser, which works by lowering the air pressure inside the vessel, causing CO2 to more readily come out of the solution. You can also use a wine whip, which is made to agitate the wine below the surface without causing excess oxygen exposure.
Do not ever shake your mead to degas. This will usually result in a messy rapid-degassing that causes lids to go flying, geysers, and sticky messes.
Take special care when making a mead with fruit or other heavy/bulky additions - like flower petals or herbs - in the fermenter, this mass will form a layer on top of the mead called a cap, which allows even less CO2 escape the solution. For such meads, it is necessary to break the cap, punch it down, and degas often. If you don't, the cap can be pushed out of the fermenter by the gasses building beneath, and the high concentration of CO2 will stress your yeast, causing off flavors.