The Temple lighting versus Chanukah lights:

A. The Temple Lighting-Seven candles lit by night and one by day:[1]

All seven candles of the Menorah were lit at night. By day, only the 6^th[2] [and 7^th[3]] candle was lit.[4] This candle was called the Ner Hamaravi. The reason for this is as follows: Night represents a time of spiritual darkness, a time for evil forces to come out and wander. At this time, it is most necessary to have spiritual light come out and shine the darkness, hence banishing the evil. During the day, however, there is not much need for spiritual light, and hence only one candle was lit. On a deeper scale, this is explained as follows: The seven candles of the Menorah represent the seven Middos of Kedusha, from Chesed until Malchus. Now, these seven Middos also exist in Kelipa; such as the Middah of love of G-d in Kelipa exists as a Middah for physical lusts, and a fear of G-d in Kelipa exists as a fear of foreign items, as well as anger. Through lighting the seven candles at night one transforms these Middos of Kelipah to Kedusha. Light comes from Chochma, and it is the light/Chochma of the candles that refines the evil Middos.

B. The Temple Menorah was lit by the right side; The Chanukah Menorah by the left side:[5]

In times of the Temple, the Menorah was lit inside the Kadosh, by the southern wall. In the directions of the Temple, East is referred to as front while west refers to the back. This would make the South be considered the right and the North the left. It thus ends up that the Menorah was lit by the right side. This is in contrast to today, that the Menorah is lit on the left, opposite the Mezuzah. The reason for this is as follows: In times of the Temple, the Menorah contained seven branches, corresponding to the G-dly light of Seder Hishtalshlus. This Divine revelation is defined and limited and can be nurtured by the Kelipos, fueling their evil activities. Therefore, the Menorah of the Temple was not lit by the left side, as the left represents Gevurah which gives place for Kelipa, and lighting in that direction could nurture the Kelipos. However, the Chanukah Menorah contains eight candles, representing the Divine light that is above Seder Hishtalshlus. This Divine light not only cannot be used to nurture the Kelipos, but on the contrary, blinds and eradicates them. Therefore, the Chanukah candles are lit by the left side, in order to refine the left, refine the Kelipos, and elevate them towards Kedusha.

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[1] Torah Or Vayeishev p. 29a

[2] Shaareiy Orah p. 6 writes only one candle was lit

[3] See Rashi Menachos 86b

[4] This follows the opinion of Rashi Menachos 86b; Tosafus ibid; Ramban; Ravad; Rashba; Implication of dialect of Avoda of Yossi Ben Yossi Kohen Gadol, printed in Seder Avoda of Yom Kippur in Siddur Arizal and Admur [letter in Likkutei Sichos 23:354]

[5] Mamar Parshas Mikeitz 1988 “Veata Berachamecha Harabim” [Printed in Melukat Daled p. 102]