Testing is a process to check whether the software meets the user requirements or not.
During software development, the developers make mistakes (un-intentionally) at different stages. So, testing helps in correcting those mistakes.
Testing also helps to identify, if the user requirements have been captured correctly. Usually, errors / issues in user requirements are captured at an early stage ie. during the preparation of SRS document to ensure that we have captured the user requirements correctly.
However, testing is generally considered as a separate phase that starts after software construction phase.
A software is said to fail if the actual results do not match with the expected results.