There is a concept called hidur mitzva, which means that we enhance mitzvos we do to make them beautiful. Examples of this principle include using beautiful esrogim on Succos, using larger tefillin and arranging for a megillah to be written by the best scribe.
The basic mitzvah of Chanukah is that the householder will light one candle each night on behalf for all the residents. The next stage is where another candle is progressively lit as the holiday progresses. The ideal method of performance is where each resident lights progressively
The Brisker Rav quotes the Rambam as codifying the act of lighting in the singular, indicating his view that there is no such step as the final one mentioned above, and that therefore the best mitzvah one can do is for the householder (but not each member of the house) to light progressively, which Sefardi Jews do.
This is at odds with the Rema, whom Ashkenazi Jews tend to follow, who maintains that each person lighting is ideal.
What is the disagreement over?
The Gemara in Shabbos discusses a Bris Milah, where the Mohel realises afterwards that he has left a small piece of skin. There are two possibilities with this surgical error; one that leaves the baby considered uncircumcised, and the other does not matter, meaning the mitzvah has been fulfilled. The Gemara concludes that there is no need for the Mohel to repeat the Bris if it is the type which does not matter.
Rashi explains that it is only when the circumcision takes place on Shabbos that the Mohel does not return, but that on weekdays he would. The Rambam disagrees, and says the Mohel would not perform the operation again even on a weekday.
The Brisker Rav sheds light on the issue: after the time of the mitzvah has past, the mitzvah cannot be improved. There is no doubt that this is the case on Shabbos, where there is universal agreement that one does not break it for the hidur of removing the leftover skin, but the Rambam says that once the Mohel has finished the Bris, he cannot make it any more beautiful than it was, as the mitzvah has been completed and therefore gone.
The Rema and Rashi disagree, and say that yes, you can! This is the same difference with regard to lighting menorahs. The Rambam says that once the householder has lit, there is no further possibility for the rest of the household to perform a hidur, as the basic mitzva was already completed when the householder had lit the first light, so the hidur stops once he has lit additional lights. Any further attempts at beautification by doing more, eg everyone else lighting, are after the mitzva has passed, so are redundant.
Ashkenazim follow the opinion the Rema and Rashi, that we can enhance something after the main mitzvah has been completed, which is why each of us lights our own menorah.