Incense can be made of a variety of things.
The low quality stuff is more often than not just enough charcoal (charred wood type of charcoal, most often) to sustain an ember. The scent in the cheap stuff is often a synthetic scent oil or powder, poured over or mixed in the medium.
Higher quality incense has a better charcoal content, and will have more natural aromatic ingredients, even if those ingredients happen to be essential oils in some cases. But more often, wood scents (such as sandalwood) will be actual powdered wood; herbal or spice scents will be the actual herbs or spices, floral scents will be dehydrated and powdered petals (though this is also where some essential oils will come into play), and there may be actual resins mixed with the medium.
So what is incense and how is it made?
Stick and cone incense is made when all of the materials are mixed together with water to form a paste called dhoop in India. I do not know the different terms for the stick type, but cone incense from India is often called dhoop cones, as it is formed of just the paste in stead of being applied around a stick. The paste is formed, and allowed to dry thoroughly. It has enough of a base medium to sustain an ember and slowly burn the aromatic component.
There are, of course, other types of incense. There is a powdered type, made to sustain an ember, but it remains a powder. These are used mainly in dishes full of sand or ash. A stamp is sometimes used, but no matter; a trench is created for the incense powder. One end is ignited, and the ember slowly burns to the other end.
Another type of incense is loose incense, which does not sustain its own ember. It is added in small amounts atop a burning coal. The mass-produced coals which are called self-igniting are impregnated with an ignition agent, typical saltpeter. It is not safe to breathe the smoke from this during the lighting process. Some people will use am ember from a fire, or something like bamboo charcoal. The latter options are either more involved to acquire (having a fire with quality of fuel and time burning enough to have embers to use) or more expensive (in the case of bamboo charcoal) than the self-igniting discs.
One more type of incense is simply using a dry stick of aromatic wood, such as palo santo or sandalwood. An end is lit, typically with a candle, and the stick is tilted to establish an ember. The not-lit end is pressed into a base, such as a small dish of sand, and the ember slowly smoulders down the wood.