If you look up the word enquiry in the online Longman Dictionary of Contemporary English, you’ll see no definition but the following remark ‘especially British English another spelling of inquiry‘. To find out what ‘enquiry’ means, you need to click on ‘inquiry’ in bold type.
This is another curious case. Both enquiry and inquiry can mean question, inquest or investigation and be used interchangeably. However, in the UK there’s a slowly growing tendency to use inquiry to refer to the official process to find out about something,
e.g. The villagers are demanding a public inquiry into the fire that destroyed the local school.
whereas the use of enquiry is becoming preferable to mean a question as in
I’d like to make an enquiry about your language courses.
One day enquiry and inquiry are likely to start living their separate lives, but until then you can use whichever you prefer (inquiry is more widely accepted in American English).
Now you know!