There are different authorities that state the chanukah menora should be lit from the left or the right. Which is correct?
20th c. rabbi and legal decisor Yisrael Meir Kagan wrote in the Mishna B'rura (Orach Chayim 676) that the guiding principle in lighting the candles is never to pass one's hand past a candle without lighting it since that would constitute passing up an opportunity to perform a mitzva. So, assuming a right-handed person standing slightly to the left of and facing a m'nora, one would light the leftmost candle first since it is closest, then proceed to the right.
Despite the fact that everyone seems to assume you light from left to right (don't take my word for it; try a simple Google search!) there actually is a difference of opinion.
The following information is taken directly from Halachipedia (with some very minor editing by me for formatting purposes):
Maharik (Responsa 183, cited by Beit Yosef 676:5) writes that on the first night, one should light the rightmost candle and on subsequent nights should add a candle to the left and light the new one first such that one lights from left to right (the way English is written). He bases his argument on the Gemara (Sotah 15b) that a person always should turn to the right, which the Mordechai (Shabbat 2:268) applied to lighting chanuka candles. The Shulchan Aruch 676:5 codifies this as halacha. This is also the opinion of the Arizal (Shaar Kavanot pg 108c), Nagid VeMitzvah (26:72), Maharil (quoted by the Magan Avraham 676:5).
The Trumat Hadeshen 106 agrees that if one is lighting opposite the Mezuzah then one should light from left to right with the new candle is always the leftmost candle which is within a Tefach of the door. However, if there’s no mezuzah and one is lighting on the right side of the door as one enters then one should light right to left so that the new candle is always the rightmost candle and is within a Tefach of the door. The Sh”t Maharshal 85 agrees with the Trumat HaDeshen. However, the Bet Yosef 676:5 quotes the Trumat HaDeshen and argues that there shouldn’t be any difference whether one is lighting on the left or right of the door one should always light the new candle first and light from left to right.
However, the Levush (676:5) and Taz (676:6), however, argue that the Gemara means in one’s first decision between right and left one should go right, but afterwards one may continue to follow that path even if that means going left. Therefore, they rule that on the first night, the candle is placed in the leftmost position, and on the subsequent nights, the candles are put to the right of the previous candles and are lit from right to left. This is also the opinion of the Sh”t Panim Meirot 1:98 and Sh”t Semach Tzedek O”C 67.
A third approach is that of the Gr”a (Bei’ur HaGra 676:5 and Maaseh Rav 240). He writes that one always should light the candle closest to the door first, even if it is not the newest candle and even if it means lighting from right to left. This is also recorded in Maaseh Rav (Siman 240).
Halacha: Mishna Brurah 676:9 quotes the Bet Yosef and the Gra and concludes one can do like either one. The Pri HaChadash, Bear Sheva (Sotah 15b), Nezirut Shimshon (Sotah 15b), Sh”t Chatam Sofer O”C 187, Chazon Ovadiah (Chanuka pg 33) argue on the Levush and hold like S”A. Kovetz Hamoedim (Moriah pg 61), Evan Israel (9 pg 129a), Sadeh HaAretz O”C 3:33, and Nehar Mitzrayim Chanuka 7 argue on the Gra and hold like S”A. The Kitzur S”A 139:11, Kaf HaChaim 676:31, Aruch HaShulchan 676:11, Natai Gavriel (Chanuka 28:2, pg 177), and Yalkut Yosef (Moadim pg 229) write that the halacha and minhag follow Shulchan Aruch. Rav Mordechai Willig (Hilchos Chanuka and Purim #1, 37-8) observed that the minhag is like the S”A.
Rav Hershel Schachter (Halachipedia Article 5773 #10) said that common practice is to put the candles in from right to left. He explained that the idea is to start the candles within a tefach of the doorway.
For visual application, see this photo from Nitei Gavriel (Chanukah pg 637)::