What's a 'four alarm fire'?Is there a more 'alarm' fire? Are there 'three', 'two', 'one and 'no alarm' fires too?

I think that is the number of firefighters called in. The bigger number alarm = more firefighters.

Now I looked it up. Churches used to ring the bell a certain number of times to alert the fire department as to how many firefighters they needed. The more rings (alarms), the more firefighters needed. Interesting.

See here

In Britain the fire brigade sometimes say that they called [number] of appliances, when describing the gravity of the fire (appliances? What happened to 'fire engines'?)

Church bells, huh? When I was a boy in this village in the 1950s, the fire brigade were directed to the fires by the sounding of the air- raid siren sounding the 'all clear', a constant steady noise. Every village had, and has, a siren because of bombing raids in WW2. The brigade steered towards the noise The commonest fires in those days were of the thatched rooves on the cottages catching fire. Nowadays we still have the cottages but the thatch is treated and there aren't so many open fires burning material that sends sparks up the chimney.

Thanks for the answer

Here's a nice article on the subject. Looks like it's from a New York source.

In my experience, the number of "alarms" is the number of times you call in more backup. It is the number of firefighters, but that doesn't equate to the same number or amount of equipment from municipality to municipality. Rather, you have the first alarm and then need more resources so you call for a second, or third, or however much more backup you need.