Does one need to Daven Marriv before lighting the Chanukah Menorah or can you light before davening? Why?
Mishnah Berurah (672:1, and in Beur Halachah ibid. ד"ה לא מאחרים) says that it depends:
If you usually daven Maariv before nightfall, then prepare the menorah ahead of time, and light immediately after Maariv (so you don't miss the proper time for lighting).
If you usually daven it after nightfall, then it's better to light first and then daven.
In the latter case, if you didn't have the chance to light before nightfall, then daven Maariv first (since it is a more usual mitzvah, so the rule of תדיר ושאינו תדיר applies; plus it includes the Biblical mitzvah of reciting Shema), and then light.