Fire is a reaction between molecules in gases. It may look as if a piece of wood is burning, but actually the burning happens in gases given off by the wood as it is heated.
Burning wood, paper etc is a complicated business, so let's take a relatively simple system like burning the gas in your cooker (assuming you use a gas and not electric oven). Actually even reacting gas (methane) with oxygen is a multistep reaction, but basically a methane molecule and oxygen molecules collide, react, and the reaction products split apart with more speed than they started with. The extra speed of the molecules comes from the energy liberated in the reaction.
So the products of combustion are gas molecules (mainly H$_2$O and CO$_2$) moving at high speeds, and in a gas the speed of the molecules is related to the temperature. High speeds mean high temperatures. In other words a flame is just a hot gas.
But when we think of fire we think of the glowing flames. The glow comes from two sources. By far the most common source of the glow is when the molecules containing carbon do not fully burn but leave behind tiny particles of carbon i.e. soot. The particles are heated by the hot gas and they glow just as anything glows when it gets very hot. The yellow/red colour of flames is due to these glowing particles of soot.
The other source of the light happens because the energy (i.e. the temperature) of the combustion products is not all the same. At any temperature the energy of gas molecules is distributed according to the Maxwell-Boltzmann distribution and a small fraction of the gas molecules can have exceedingly high energies. High enough in fact to cause ionisation of other gas molecules, and light emission as they recombine.
It's this ionisation that leads some people to talk about a flame as a plasma. However you need to bear in mind that only a tiny tiny fraction of the combustion products are ionised, so it isn't a plasma in the sense that the Sun contains plasma.