A test fails because an actual result was not expected, so...
ideally, the failure is a symptom of a software (AUT) fault
or it may be a symptom of a test fault
or it may represent a fault in both the test and the AUT
which would imply a fault in the specification
which both testers and developers (and tech writers) 'assumed' they understood
or ... least likely but possible, there was a glitch in the test environment.
So the first tasks following failure are to document the observation and to attempt fault isolation. (Attempting to reproduce the symptom is part of fault isolation.) Severity can be assessed immediately.
Defects become part of the backlog, and so are prioritized amid everything else; let us not fall into the trap of drifting from our backlog of work items to focus on defects. Let us capture the Severity of the defect without letting that severity fully determine the priority of the defect. This applies whether you create a triply distinct capture as suggested by Testing Computer Software or follow a business impact-sensitive severity classification later advocated by Kaner and written into IEEE 1044. Especially (but not exclusively) as the product nears delivery, it may make more sense to work minor or even cosmetic severity defects on Must-have functionality than to repair Critical issues in Would-like functions.
Formal definitions of Failure and Fault are at this link, although I no longer agree with writing down great detail of every specification (for software) and believe successful agile methods provide empirical proof of lack of need for such detail. For an excellent formal example of noted ambiguity, look at this definition of Error. There is a formal definition of glitch as a brief unintended voltage on a transmission wire, but I'm using the informal meaning here.
AUT stands for "Application Under Test' which is part of the SUT or "System Under Test." Thanks to HASI/GEADGE testing for teaching, and Ron Damer for reinforcing, the simple, obvious idea that "system" testing is always more than software.