What the vast majority of people would probably think of when they hear the word 'fire' is actually flames; flames are quite simply particles emitting light.
For an everyday example, take a campfire: The wood logs you see burning are at such a high temperature that they give off methanol (and other flammable chemicals), which is most of what's burning. Apart from the methanol being driven off of the wood, there will be other chemical compounds and/or larger clumps of more-or-less-burned wood that will be carried off. These larger clumps in particular, while very small, are nonetheless large and hot enough to start emitting light in the visual spectrum. This is essentially what a flame is: Particles emitting light.
Fire in and of itself is quite simply rapid oxidation in the presence of oxygen.
The above is, arguably, a gross oversimplification.
TLDR: Flames are particles, fire is combustion.
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