A grievance is a complaint. It can be formal, as when an employee files a grievance because of unsafe working conditions, or more of an emotional matter, like a grievance against an old friend who betrayed you.
A grievance is a complaint that may or may not be justified. Often a grievance is a complaint about something that breaks a rule or a law, like a grievance filed against a person who builds a fence in his front yard in a town that does not allow that. If that same neighbor plants flowers he knows make you sneeze uncontrollably, you might be unhappy about that, but this is the kind of grievance you can work out among yourselves.