And you shall command the people of Israel, that they bring you pure oil olive for the light, for the lamp to burn always. (Exodus 27:20)
The Shulchan Aruch (Code of Jewish Law) prohibits specific oils for use in kindling the Shabbat lights, such as those derived from animal fats and fish entrails, but permits all other types of oil. However, it concludes that while olive oil is not essential, it is certainly the most desirable oil for performing the mitzvah of kindling lights for Shabbat (Orach Chaim 264:6).
The Talmud itself explains that while the sages may have permitted a range of oils, olive oil was the best as it produces a clear bright flame (Shabbat 23a). Rabbi Yisrael Meir HaKohen Kagen (d. 1933) explained that olive oil is also drawn through the wick more effectively than other oils (Mishnah Berurah 264:23).
Rabbi Moshe Feinstein (d. 1986) states that modern wax candles are equally as effective at producing a clear flame and are therefore equivalent to using olive oil. The only exception is with the festival of Chanukah where it is preferable, but not essential to use olive oil to help remind us that the miracle was performed with olive oil (Dibrot Moshe 469:22). The Mishnah Berurah adds that since it is forbidden to tilt a flame on Shabbat to brighten it, candles which do not need to be tilted to improve their light are in fact preferable to all oils – even olive oil – on the grounds that one will never be tempted to break Shabbat to improve illumination (Mishnah Berurah ibid.).
Nevertheless, many still follow the custom of the the Vilna Gaon (d. 1797) and only used olive oil for Shabbat lights. The reason may be based on a Midrash which explains that the teaching of olive oil was repeated regularly because of the special love God has for olive oil (Tanchuma on Behaalotecha, see also Talmud Shabbat 26a).
Either way, the idea of kindling Shabbat lights is to illuminate the Jewish home on the most special and spiritual time of the week. This way, families can enjoy their Shabbat together and continue to strengthen the bond between them.