Incense is any organic material that releases fragrant smoke when burned or heated. The most common types of incense are all tree products, particularly the aromatic barks and seeds of certain trees (think cinnamon) and the oils and resins that can be extracted from them. Therefore incense tends to be made out of more or less the same stuff wherever you go, even though the particular plant you're burning might not be the same from place to place.

By far the most common kind of incense used in the ancient Mediterranean was frankincense, which was so common that ancient texts will often just call it incense as if there's no other kind. Although frankincense bushes also occur naturally in India and parts of east Africa, the Greeks and Romans generally got theirs from Arabia (which, I mean, is a lot closer...)--it does not grow in the Mediterranean, and Pliny mentions that he doesn't actually know what the bushes look like, saying that even Roman expeditions into Arabia brought back no clear consensus on what exactly the bush looked like. It was used quite commonly in ceremonies, burned at altars, etc. The Greeks burned it at weddings and symposia and the burning of incense was often a part of the ritual of mystery cults. Plutarch tells the famous story of Alexander's tutor Leonidas scolded him as a boy for throwing to handfuls of frankincense into the fire during a sacrifice, and told him that until he conquered the land where frankincense came from he would have to be a little bit more sparing--when Alexander conquered Gaza, says Plutarch, the major distribution center of incense along the "Incense Road" from Arabia, he sent Leonidas a gift of five hundred talents' weight of frankincense and a hundred of myrrh, with a note that Leonidas should be a little more lavish with his offerings to the gods. The Romans mainly traded for incense by sea, rather than the land route the Greeks used, and it was common in almost all their sacrifices, in cults of the dead, and in imperial worship, as well as in mystery cults borrowed or adapted from the Greeks or elsewhere (e.g. Cybele or the cult of Isis). Cato mentions including a small amount of frankincense in medicine for cattle and Pliny mentions it being used medicinally as well.

Myrrh was also a common incense, but doesn't seem to usually have been burned. Also from Arabia, myrrh was mostly used as a spice and a medicinal element--the New Pauly says it has "anaesthetic, warming and astringent effects," and mentions Dioscurides prescribing it for coughs and respiratory problems. It's mentioned as being used in ointments (perfumed unguents are among our most common references in elegaic poetry) and to spice wine.

Terebinth is, to my knowledge, the only other major type of incense for the Greeks and Romans, although I know cedar was burned by certain Near Eastern cults. This also wasn't burned, although it is now--its resin was used in poultices or was ingested medicinally, and its wood was used to make cups and stuff