A chapel is a church or area of worship. Sometimes it is small and attached to a larger institution (a cathedral or large church, school, hospital, prison, airport etc). You can find small chapels inside churches and cathedrals, sometimes dedicated to the family that paid for their construction.
A church is a place of public (usually Christian) worship, usually larger than a chapel. The term "church" also refers to an organisation of people who share the same beliefs (eg the Catholic church).
A cathedral is a really big church, and usually the main church in a particular diocese. All cities in England have a cathedral, whereas smaller towns don't.
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