Hanukkah candles are generally lit once it is dark outside each night of the holiday. The main exception is Friday night, when they are lit before Shabbat candles, before sunset.
Usual time each night
Most traditional practices light the menorah sometime between sunset and full nightfall, so that the candles will be visible in the dark and burn for at least about 30 minutes into the evening. Different communities follow slightly different exact times (right after sunset, some minutes after, or at nightfall), but all do so after sundown on regular weeknights of Hanukkah.
Friday night of Hanukkah
On Friday night, lighting a new fire after Shabbat begins is not permitted, so the Hanukkah candles are lit before Shabbat candles and before sunset. People usually use longer candles or more oil so the lights will keep burning well into the evening, since they are lit earlier than usual.
Saturday night of Hanukkah
On Saturday night, the candles are lit after Shabbat ends, once it is dark again. Communities differ on whether to do the havdalah ritual first or to light the menorah first, so many people follow the custom of their own family or community.