Automation Testing is use of tools to execute test cases whereas manual testing requires human intervention for test execution.
Automation Testing saves time, cost and manpower. Once recorded, it's easier to run an automated test suite when compared to manual testing which will require skilled labor.
Any type of application can be tested manually but automated testing is recommended only for stable systems and is mostly used for regression testing. Also, certain testing types like ad-hoc and monkey testing are more suited for manual execution.
Manual testing can be become repetitive and boring. On the contrary, the boring part of executing same test cases time and again, is handled by automation software in automation testing.