Smoke testing
A smoke test is a short, high-priority check of essential paths. It answers one question: is this build testable?
Regression testing
Regression testing is broader. It checks whether changes have damaged features that already worked.
When to use each
Run smoke tests immediately after deployment. Run regression testing after the build passes smoke testing and before release.
A practical example
For an online shop, smoke testing might cover login, search and checkout. Regression testing might also cover discounts, refunds, stock and notifications.