Both words mean the same thing; the only difference is where you are writing. Labor is standard American English, while labour is standard British English. A quick memory hook: British spellings tend to keep the extra u, as in colour and honour, and labour follows the same pattern.
The core difference
Labor and labour are the same word, spelled the American way and the British way.
- labor — the American spelling, used for work, effort, or workers as a group: Manual labor built the bridge.
- labour — the British spelling, used the same way, plus the process of childbirth: She was in labour for fourteen hours.
How to tell them apart
There is no difference in the core meaning between labor and labour. The split is purely regional: American English dropped the u from a group of French-derived words, while British English kept it, so labour sits alongside colour, favour, and honour. If you are writing for a British audience, or referring to the British political party Labour, keep the u. For American readers, or terms like the U.S. Department of Labor, drop it.