Image: My chanukiah, 2nd night of Chanukah. Photo by Rabbi Ruth Adar.
The nine-branched lamp we use at Chanukah is most correctly called a chanukiah, (pronounced khah-noo-KEE-ah.) It is a calendar of a sort, measuring out the eight days of the holiday, with an extra candle to light the others.
However, you will often hear people refer to one as a menorah (muh-NOH-rah or meh-noh-RAH). That is also correct: in Modern Hebrew, menorah means lamp. The thing on your nightstand is also a menorah, but it is unlikely to work as a calendar.
How to light the chanukiah/menorah:
May your holiday be bright and warm!