People in this play talk about "my lawsuit" just as easily as other people might say, "my pencil" or "my piece of paper." It's nothing to them, and everyone who's anyone has one. We tend to think of lawsuits as a pretty big deal. They're expensive, and it means that someone is super mad at someone else.
But in Moliére's world, lawsuits are a way for aristocratic men (and women) to settle scores against each other. In some way, you could think of them as a civilized answer to a duel. You know, someone insults your poetry; you slap them across the face with a glove and tell them to name a second and meet you at dawn.
But not in the court of Louis XIV. If you had any class, you wouldn't pull out a glove; you'd pull out your lawyer and file a suit. In other words, lawsuits enforce society's rules, not any sort of legal rules that we'd recognize. Check out the way Alceste refuses to play according to the rules: "No, I refuse to lift a hand. That's flat./ I'm either right, or wrong" (1.1.96). And the result? Just as Alceste's poor social skills have excluded him from society, losing his lawsuit literally banishes him from Paris. It's as if he'd refused to duel.
Célimène, on the other hand, knows how to win life and her lawsuit because she's awesome at manipulating people. When Alceste asks why she hangs out with Clitandre, she tells him, "My lawsuit's very shortly to be tried,/ And I must have his influence on my side" (2.1.45).
How well does it work? Well, no one comes knocking on her door to drag her down to the station.