Why do some people have the custom to wind the Tefillin-Shel-Yad towards themselves, while others wind it away from themselves? What is the reason behind these two customs, and which sects of Judaism generally follow which method?
It's pretty much a question of which eida you come from:
Ashkenazim wind inwards, Sfaradim wind outwards.
(And, confusingly, right-handed ashkenazi using left-handed ashkenazi tfillin would also be winding backwards...)
The link that @jake posted in comments illustrates this very nicely - it depends on which eida's minhag you use.
(Sorry, I don't know the reasoning behind the different minhagim - though I find it likely that each do it that way, because thats the way they do it...)
Ashkenazim wrap the strap of the tefillin shel yad inwards. The reason is that this constitutes what Devarim 11:18 says concerning binding the words on one's heart and soul, as well as one's arm and between one's eyes.
However, Sephardim and Hasidim (with exception to Gur) wrap the strap of the tefillin shel yad outwards. This is to remind the wearer of the tefillin of the common kabbalistic dictum that one must give before they can receive. When wrapping the strap outward, the wearer brings the strap away from himself before wrapping around his arm and coming back towards himself.