The plaintiff is the person who files a law suit. The defendant is the person being sued.
It is extremely important to be able to tell who the plaintiff and who the defendant are in the opinions that you read. The first question that almost all law school professors ask the first day of class is: "Mr. X, who is the plaintiff in Noone v. Someone?"
Sometimes, the correct answer is, "You can't tell what the names of the plaintiff or defendant in this case, Professor Z. Even though most courts put the name of the plaintiff first, some courts switch the order of the plaintiff and defendant when a case is appealed. Thus, sometimes, the defendant's name might be first in the style. But not all courts follow this convention; the name swapping convention created confusion so most courts no longer use it. Now, the plaintiff's name is usually always first.
But your professors really aren't looking for the names of the plaintiff and the defendant when they ask you who they are. They want you to explain to the class how the particular players are positioned in the suit.
In this case a good answer would be: "The plaintiff in this case is the non-marital partner of the defendant doctor. The plaintiff is claiming that the defendant doctor broke her promise to pay."