In a wood fire (and also similarly in a candle flame, or the flame of a simple lighter, or a bunsen burner with the air valve closed, number 1), because the air has to flow into the flame from the outside, there is not enough oxygen in the flame to burn all of the fuel.

Because the fuel in all of these cases is a carbon-based material, one of the products of this incomplete combustion is just... leftover solid carbon, which forms into tiny particles of soot. And because these tiny particles are in a hot flame, they are also hot, and so glow a bright yellow.

On the other hand, in a gas burner for heating, or a bunsen burner with the air valve open, the gas is mixed with air before it enters the flame. That way, there is enough oxygen available to quickly react all of the carbon and hydrogen in the fuel to carbon dioxide and water (this is what is meant by "complete combustion), so there is no soot that can glow, and you get both a hotter flame, and also more heat per each unit of fuel.

There is a perfect ratio of fuel to air that is perfect. For example, 2 parts fuel to one part air. If you stray from that perfect mixture, you'll have different flame properties

A blue flame means the fuel (gas) is combusting fully - complete. Incomplete means not all the gas is being consumed in the reaction with the excess being released into the atmosphere.