The colour of a flame is determined by solving the quantum mechanical equations governing the energy levels of a specific substance. In general when heat is supplied to a substance, it will cause electrons to be excited to high energy levels. When the electrons return to their ground energy state the release the energy in the form of radiation. It is not entirely true that most flames are red, some are blue others green, some substances burn and release radiation which is invisible. The exact colour of the radiation being emitted depends how high the electrons were excited and how low is their ground energy state.

The color of a flame is not determined by solving an equation. The equation simply lets us predict the color of the flame. Essentially, when heat is applied to a molecule, the electrons become excited and some of them can move up to higher energy levels. (excited states) These levels are in distinct places for each type of molecule. When electrons reach these excited states, they have to fall back down to their original (ground state) level. When they do this, they give off energy in the form of light. This light can be in any wavelength, depending on the distance between the two energy levels. If a fall happens to emit in a wavelength in the visible range, we see it as flame. (The flame itself is hot particles floating upwards. A glowing coal is doing the same process, but it's not moving) As I said before, different molecules have different energy levels that their electrons can go to. This produces different wavelengths, and thus different colored flames. Carbon, at reasonable temperatures, produces a lot of orange, and so we typically think of fire as orange or red. This process creates energy in wavelengths other than the visible range, and we can sometimes feel this as heat. This phenomena is used in chemistry to determine the chemical makeup of a compound by analyzing the different types of wavelengths in the IR or UV and visible ranges.

The color of a flame is determined by certain metals that burn. (Different metals are put into fireworks to make them different colors)

in short, the temperature. and, i don't agree with you about them usually being red.

The heat of the flame. The higher the heat the brighter the flame. Red and orange are actually cold heat. While blue, green, and white are the hottest.