A hanukiah, AKA a chanukkiyah or 'Hanukah menorah', is the special, nine-branched menorah used during the celebration of Hanukah. Goyim and non-Israelite Jews alike call the hanukiah a menorah, and there is nothing wrong with doing so. But in the original instructions on how to make the 'official' menorah for the Temple in Jerusalem (Exodus 25:31-40), it was explicitly specified that the menorah should have seven branches.
The hanukiah has nine branches, two more than the original menorah. One branch, usually the central branch, is called the shamash and holds that candle that is used to light the other candles. While in the original menorah it was the central branch that held the shamash, many hanukiah will have the shamash on the far right. The eight remaining branches represent the eight days of Hanukah. On the first day of Hanukah one candle (in addition to the shamash) is lit, on the second day, two, and so forth.
Aside from the symbolic importance of the eight candles representing the eight days of Hanukah, a hanukiah (or other menorah with some number of arms other than seven) might be used because in halakha tradition it is forbidden to make a menorah in the exact form of the original. This tradition has a lot of leeway, and it is not uncommon to see seven branched menorah.
The word 'hanukiah' appeared in the late 1800s, during the revival of the Hebrew language, and is in common use in Israel. It has not yet spread to the rest of the world, and probably never will, seeing as how the word menorah is amply fulfilling our linguistic needs. But next time you want to show off your leet Judaic vocabulary, you'll be ready.
There is a large body of tradition surrounding the use of the hanukiah. The candles should be placed in the hanukiah from right to left (the direction that Hebrew is written), one new one added each night, and then lit from left to right (the leftmost candle representing the current day). The candles should be left to burn for at least half an hour, and are usually left to burn out on their own rather than be blown out. (TheLady reports that it is her tradition to place the candles left to right; as she uses the asymmetrical hanukiah mentioned above, this results in a more symmetrical placing of the candles).
The central candle, the shamash, is an important part of the hanukiah, because it is the only one without religious significance (it does not represent any of the days of Hanukah). Thus, while you are not supposed to use any of the other candles for any practical purpose (for light or for lighting another flame), the shamash can be used for these purposes. The hanukiah should be placed in a window or other public spot, to share the spirit of the holiday with the world and be an 'or la'goyim.' This last requirement has been waived in the past when Jews might be in danger of prosecution if found to be celebrating Hanukkah.
There are a number of blessings that should be said for Hanukah and for the lighting of the candles; I hope that someone more knowledgeable than I will node them.
The plural of hanukiah is hanukiot.