A thurible is a working vessel rather than a purely symbolic one, built to hold a lit coal at several hundred degrees while being swung on chains without spilling it, and its construction follows from that requirement.
Construction and fuel
Most thuribles are brass or silver, a bowl with a pierced, hinged lid suspended by three or four chains from a disc that lets one hand manage the chains and the lid together. A charcoal briquette, lit in advance, sits inside, and incense grains, commonly frankincense, myrrh or benzoin, are added a spoonful at a time from a small boat shaped container called a navette. Orthodox censers add small bells to the chains, so the censer rings softly as it swings, a feature Western thuribles generally lack.
Who handles it
The thurible itself is carried and swung by a server called a thurifer, a role that can be a layperson, but the incense is imposed into it, meaning placed on the coal, by the priest or deacon presiding, usually with a blessing spoken over it first. This division, server carries and swings, clergy loads and blesses, is fairly consistent across Catholic and Orthodox practice. A thurible is a utensil rather than a consecrated vessel in the same sense as a chalice, so a worn out one is simply retired and replaced rather than requiring formal disposal, though the ash is emptied respectfully rather than tipped out with household rubbish.