How and when and by who were the Others first defeated?
When was the Wall erected compared to the time of their defeat? If that can be answered, it will probably imply the answer to this: was its purpose primarily as an active defence (to defend during an ongoing conflict with the Others) or as a pre-emptive defence (to defend against them if they came again.)
A Song of Ice and Fire is interesting because its distant history is always a little untrustworthy. History from The Age of Heroes is sometimes doubted to be real and anything older is more legend than fact.
But as far as we know, Brandon the Builder made the wall after The Long Night, which was about 8000 years prior to the first book.
Once it was discovered that The Others could be killed with dragon glass, the children of the forest and mankind was able to push back and seemingly defeat The Others. The wall was then built to ensure that the realm would never be caught off guard again.
As for how it was built, that appears to be lost with history for now. We will have to see what Sam learns in Oldtown in future books, hopefully some "lost history" will be uncovered.
The time the Others were first defeated was during The Long Night. It is said that the Children of the Forest, the First men and possibly a hero called Azor Ahai defeated the Others, and (presumably) some time after, the Wall was built by Brandon the Builder, who supposedly was a Stark.
All of this took place so long ago that no one is really certain of what happened. But as we have seen, the Wall does repel undead (Coldhands could not pass the Wall, ASOS, Sam. Melisandre is impressed with the magic radiating from the Wall, ADWD, Melisandre.) So it is clearly meant as a defence against the Others and their minions.
It is mentioned to be built after the war, as a ward against the Others returning. There is some indication that humans were driven south by the undead legions, which to me implies that they had already been driven past the current location of the Wall. But we really have no clear evidence either way, since this is said to have happened before people even kept historical records (except by carving runes into rocks).
The Others last (and perhaps first; there are no tales of an earlier appearance, though it may just be forgotten) appeared about 8000 years ago and were defeated by Azor Ahai wielding the magical sword Lightbringer and leading other humans using dragonglass weapons, allied with the Children of the Forest.
Afterwards, the wall was constructed by the founder of house Stark, Brandon Stark, also known as Bran the Builder, supposedly with the help of giants. The wall was not originally built to its current height; the newly established Night's Watch kept adding to it in the following millenia.
However, all of this is the matter of legends; there are no surviving documents from that time, and the wall itself (and more recently the reappearance of the Others) is pretty much the only solid proof that it's nost all just fairy tales.
