There are a few categories of material that have their own distinct halachot when koshering utensils.
Metal, Stone, Bone, and Wood
Metal, stone, bone, and wood can all be koshered (שו”ע או”ח תנא:ח).
Cheres – Pottery
The Torah (ויקרא ו:כא) writes וכלי חרס אשר תבושל בו ישבר – pottery which the Bnei Yisrael captured in war must be broken. The Gemara (פסחים ל:) writes that the Torah is saying that pottery can never be koshered. The Gemara continues, an oven may be koshered for Pesach, but pottery must be destroyed. The Gemara explains that even though ovens are usually pottery, this oven was metal and therefore it may be koshered. Alternatively, the oven was pottery, but the oven gets heated on the inside where the food was, as opposed to a utensil for food which gets heated from the outside, the opposite side of where the food touched the utensil. The Gemara adds, even if one put coals inside a utensil of pottery, it would not help. This is because we are concerned that the person doing the libun will get nervous that the utensil will break from the intense heat and remove the coals before the utensil reaches the temperature of libun. However, the Gemara adds, if you put the pottery into a furnace, since it is so hot that it reaches libun temperature almost instantly, there is no concern that the person will remove the utensil early (רא”ש פסחים ב:ז). The reason that a furnace helps to kosher pottery, despite the Torah writing it must be destroyed, is because once the utensil goes into a furnace it is as if it is a new utensil.
Included in pottery is porcelain (משנה ברורה תנא:קסג) and ceramic (שולחן הלוי ח”א פרק כד אות ל ס”ק ג).
There are three opinions in the Rishonim about glass. Rabbeinu Yechiel’s (הובא במרדכי סימן תקעד) opinion is that glass is like pottery and cannot be koshered. The Re’ah (בדק הבית ב”ה ש”ו) and the Ohr Zarua (ח”ב סימן רנו) are of the opinion that glass is like metal. In whichever way it absorbed belio, that is how one must remove them, either with libun or hagala. The Raavya (סימן תמד) and the Rosh (פסחים ח:ב), among many other Rishonim, are of the opinion that glass does not absorb beliot at all.
The Shulchan Aruch (או”ח תנא:כו) states that glass does not have any beliot at all, as did the Raavya. The Pri Chadash (שם סעיפים כג,כו) rules the same as the Shulchan Aruch, that one only needs to wash out glass utensils between meat and milk. The Rama (או”ח תנא:כו), Chochmas Adam (כלל קכה סעיף כב) and the Aruch Hashulchan (שם סעיף נ) argue concerning the Shulchan Aruch’s ruling and conclude that ashkenazic custom is to treat glass like pottery. That means that not only does the Rama rule that glass has beliot, one cannot even kosher the glass.
The majority of contemporary Poskim write that the Rama is stringent only in regard to Pesach. However, for the rest of the year, the Rama would be lenient and not even require hagala between using a glass utensil for dairy and meat (עי’ בית לחם יהודה יו”ד קכא ס”ק י, מהרש”ם ח”ג סימן צד). There are some Poskim who are stringent and rule that one must make glass kosher through hagala, see Artscroll, The Kosher Kitchen (pp 340-341).
Plastic and Teflon
Most contemporary Poskim rule that utensils made of plastic, as well as pots that are coated in Teflon, can be koshered with hagala. It is proper to be stringent when koshering for Pesach and not to kosher such utensils, unless they were only used in a kli sheini.