The "hydration failed" error in React occurs when there is a mismatch between the HTML generated on the server and what the client tries to render. This can happen due to differences in random values, current date/time, or elements rendered conditionally based on the environment. To solve this issue, it's essential to identify the component causing the problem, check for differences in render output, look out for async operations, test server and client independently, and refactor problematic components. A common cause of the error is using a random number in the render function, which can be fixed by moving the random number to the state and updating it as a side-effect after the component has mounted and hydration is complete.