The Gemara (Shabbat 21b) says that all wicks and oils are valid for Hanukkah. Rabbi Yosef Messas (Shu”t Mayim Haim § 249), basing himself on the aforementioned Gemara, rules that an electric bulb is also permissible for use as a candle. Just like any other Hanukkah, the bulb’s filament is akin to a wick and produces a bright light. The Maharal (Ner Mitzvah, vol. 2, s.v. “Amar Rabbi Yehuda”) rules that the Hanukkah lights must be similar to those that were used in the Bet HaMikdash, namely oil and standard wicks, and therefore Rav Ovadia Yosef (Hazon Ovadia, pg. 96) disputes Rav Messas’ assertion that electric bulbs would be permissible. Another issue raised by the rabbis is that an electric light produces light indirectly (turning on a switch, causing a current to flow and then lighting the filament), whereas the Mitzvah of Hanukkah should be done by direct kindling.
Rabbi Shlomo Zalman Auerbach (Shalme Moed, pg. 200) and Rabbi Yosef Elyashiv (Kovetz Teshuvot, vol. 3, § 103) both rule that if someone does not have access to conventional oils or candles for Hanukkah, one may use and even make a blessing over an incandescent bulb (LED bulbs are invalid). Rabbi Ben Zion Abba Shaul adds that a battery in a battery-operated electric light fulfils the requirement of oil.
Summary: An electric incandescent light may be used for Hanukkah in a situation of great need, preferably without a blessing.