Antiferromagnetism always refers to same atom of nearest neighbor. However, as you can imagine there are different structure, and there will be different nearest neighbor. If any of this nearest neighbor is pointing opposite with each other, then it's antiferro.

You can find some different types in the following web site:

http://folk.uio.no/ponniahv/activity/ordering/spinfig.html

As for the reason for antiferro, it starts with the exchange interaction between spins.

Either positive(ferro) or negative(nagative) is possible.

Wikipedia will tell you the basic idea.

http://en.wikipedia.org/wiki/Exchange_interaction

There are oppositely aligned because this is the state with minimum energy (or the most stable state)

Yes, the atoms are the same, but the exchange interaction between them (whether it direct or indirect) is antiferromagnetic.

Thanks Mr. Zhiling Dun, Mr. P. Grima Gallardo and Mr. A. A. Timopheev for your valuable time and for clearing my concepts.