The traditions of Chanukah can be a little confusing. The complicated discussion of these traditions in Talmud tractate Shabbat (beginning on 21a) help clarify certain points, but in regards to the order in which the candles are to be lit says little.
What the Talmud does make clear for us is two points of the tradition that are applicable to today. First, it makes it clear that we are to increase the number of candles each night, according to Hillel, as opposed to decreasing the number of candles each night (as Shammai suggested). This is to recognize the increasing nature of the miracle of the oil for each night (one day’s supply lasted 8 days, so each night is an additional miracle).
The second thing that is clear from the Talmud is that we are not to use one Chanukah candle to light another and that we are not to use the light of the menorah for performing our daily tasks. These two prohibitions led to the creation of the shamas candle, the candle that is used to light the other candles. By using the shamas we assure that we do not use a candle designated to represent a night to light any other candles, and by placing the shamas on a different level (i.e. higher or lower) than the other candles, if the menorah happens to be the only light in the room, we can still do our tasks based on the light from the Shamas.
The standard order for lighting the candles is based on the Shulchan Aruch. As one faces the menorah, the first candle goes on the far right, and then each subsequent candle is positioned directly to the left of the one that preceded it. The lighting then goes in the opposite order, starting with the left most candle as one faces the menorah continuing to the right. The blessings should be recited after the Shamas is lit and as it is being used for lighting the other candles.
One additional tradition discussed in the Talmud is that while the requirement of lighting Chanukah candles means one menorah per household, it clearly states that for those who wish, there may be one menorah per family member. This is a wonderful way to involve everyone in the lighting of the menorah.
Ultimately, the purpose of the menorah and the lighting of candles on Chanukah is to publicize the miracle of Chanukah (therefore menorah’s should be placed near a front window, or even outside if possible). For families and individuals the act of lighting the menorah and saying the blessings can be a powerful reminder of Jewish identity and the importance of keeping the chain of tradition alive. If that goal is achieved, the order in which the candles are lit is ultimately not that important.