So really the three different basins are anaerobic, anoxic, and oxic.

Anaerobic -> no sources of oxygen (free oxygen, nitrate, etc). There's lots of things this does, but you're probably asking about it in terms of enhanced biological phosphorus removal. This selects a group of heterotrophs called PAO's that you use to accumulate polyphosphate later in the process.

Anxoic -> no free oxygen, but you will have other sources of oxygen (such as nitrate). This is used for denitrification, as the lack of free oxygen means your nitrate and nitrite can be used as an electron acceptor.

Oxic -> This will have both heterotrophs treating BOD (huge community of bacteria) and autotrophs nitrifying (smaller community - primarily nitrosonomas and nitrobacter.) At this stage your PAO's are also accumulating polyphosphate.

So those are the three basic stages, and there's about 50 different processes that combine them in different orders in order to achieve treatment. One thing to note is that you can ONLY denitrify after you nitrify, let your oxic basin (which is your nitrifying basin) typically comes after your anoxic basin (which is your denitrifying basin.) The reason for this is because recycling a huge portion of the flow from your oxic basin up to the front is going to save a lot of energy for a variety of reasons.

This is also why you can have 4 or 5 stage bardenpho's. You can put a post-anoxic and post-oxic tank after your oxic tank in order to improve your effluent quality, since your recycle and tank size would exponentially as you get to lower and lower TN limits. So to get around this, you do most of the denirifying in your first anoxic basin (thus saving energy), and just finish it off in your post-anoxic