Center and centre mean exactly the same thing — the middle point of something, or a place set up for a particular activity. Center is the American spelling; centre is the British, Canadian, Australian and Irish spelling. Pick the one that matches the English you're writing in and use it consistently.
The core difference
They are the same word with the same meanings — only the spelling changes by region.
- center — American spelling: the middle point of something, or a building for a specific purpose: She placed the vase in the center of the table.
- centre — British and Commonwealth spelling of the same word: The leisure centre is open until 10pm.
How to tell them apart
Both spellings come from Latin centrum, itself from Greek kentron, "a sharp point, the fixed point of a compass". American English settled on the -er ending, while British and Commonwealth English kept the older -re ending — the same pattern as theatre and metre.
The confusion usually comes up in business names, addresses and software copy that mixes house styles, for instance a British company writing "shopping center" by accident. The fix is simple: decide which variety of English the piece is written in, then keep that spelling for every instance, including in compounds like city centre or call center.