Shabbos, Chapter 2, Mishnah 1. This mishnah and several of the ones that follow it deal with the mitzvah of lighting the lamp on Erev Shabbos. Our mishnah opens by listing the types of materials that may be used as wicks and the types that may not, as well as the types of fuel that may be used for lighting and the types that may not.
The reason for the prohibition, as explained in the Gemara:
Regarding the wicks: With the kinds of wicks listed in the mishnah the flame does not catch properly; it seems to skip over them, and the burning is not steady. Since the burning is not steady, there is a concern that a person may come to tilt the wick on Shabbos and adjust it in order to get a better flame, and he would then be transgressing the prohibition of kindling on Shabbos.
Regarding the oils: There are oils that are not drawn up into the wick properly, and the burning is not stable. Here too a person is liable to come to adjust the oil so that it will be drawn up more steadily. Because of this concern, the Sages did not want a person to use these materials.
The mishnah opens: "Bameh madlikin u'vameh ein madlikin" - with which materials may one light, and with which may one not light? The term "lighting" refers both to the wicks and to the oils. The mishnah first lists the wicks.
The wicks with which one may not light:
"Lo b'lechesh" - a wick made from the material that grows in the cedar tree, between the bark and the wood.
"V'lo b'chosen" - flax that was not properly combed, and still has small, broken pieces left in it, and therefore does not light well.
"V'lo b'cholach" - an inferior type of silk.
"V'lo bipsilas she'al ha'idan" - a woolly material that grows in the willow plant, between the bark or membrane and the wood.
"V'lo bifsilas hamidbar" - a type of grass that grows in the desert from which people would make wicks. It is of poor quality, so the Sages did not want it used, out of the concern that one might thereby come to desecrate Shabbos.
And finally, seaweed that grows on the surface of the water - a green species from which people would make wicks, and it too may not be used for lighting, for the same reason.
The oils and fuels with which one may not light:
Pitch.
Wax - and this refers to liquid wax; but hard wax, of the sort used nowadays, is permitted.
Oil that comes from gourd seeds.
Shemen sereifah: This is a term describing oil of any type to which something happened that requires it to be burned. It is permitted to derive benefit from it, or at the very least others may derive benefit from it, depending on the circumstances, but it may not be used as fuel for the Shabbos lamp.
The reason for this does not lie in the quality of the burning - it can in fact be perfectly good fuel, even olive oil. The problem is a different one: our mishnah is speaking of a case where Yom Tov falls on Erev Shabbos, and the lighting of the lamp takes place on Yom Tov. Since sanctified items that became disqualified and require burning are not burned on Yom Tov, but rather one waits until after Yom Tov, this oil cannot be used on Yom Tov.
And there are those who understand that the mishnah is not speaking only of a case where Erev Shabbos falls on Yom Tov; rather, there is a problem on every Erev Shabbos, because of a decree the Sages made on account of the case where Yom Tov falls on Erev Shabbos. In other words, although by strict law the problem exists only on Yom Tov itself, according to this opinion the prohibition was extended to every Erev Shabbos by force of the decree.
The mishnah continues listing the types of fuel that are prohibited:
"V'lo b'alyah" - a type of fat found in the tail of a sheep. It is kosher for eating, and nevertheless it may not be used, because it does not burn well.
"V'lo b'chelev" - the type of fat that is forbidden to eat.
The dispute between Nachum HaMadi and the Sages:
Nachum HaMadi disagrees and says: "Madlikin b'chelev mevushal" - one may use cooked chelev. "Cooked" in this case means that it was melted into a liquid; if the chelev was melted into a liquid, one may light with it according to Nachum HaMadi.
And the Sages disagree and say: "Echad mevushal v'echad she'eino mevushal, ein madlikin bo" - whether it was melted or not melted, one may not light with it.
Seemingly this is the very view of the Tanna Kamma, who already ruled that one may not light with chelev. What, then, is the difference between the Sages and the Tanna Kamma?
The Gemara explains that one of them holds that adding a bit of kosher oil, from the oil that is permitted for lighting, renders the chelev fit for use. There is no clear resolution as to which of them holds this way, the Tanna Kamma or the Sages, but one of them holds that if a bit of oil is added to the chelev, one may light with it.