Short answers, pulled from the story.
F. S. Boas coined the term in his 1896 book Shakespeare and His Predecessors. He used it to identify three plays by Shakespeare that shifted violently in tone between comic material and dark psychological drama.
Boas originally named three plays: All's Well That Ends Well, Measure for Measure, and Troilus and Cressida. He also noted that Hamlet served as a bridge between the problem plays and Shakespeare's more clearly defined tragedies.
The term was borrowed from a type of drama closely associated with the Norwegian playwright Henrik Ibsen. In those plays, the protagonist's situation was presented as a representative instance of a contemporary social problem.
The number depends on the critic. Boas identified three; others add The Winter's Tale, Timon of Athens, and The Merchant of Venice. Ernest Schanzer argued that only Measure for Measure truly qualifies, while proposing Julius Caesar and Antony and Cleopatra as better alternatives.
Scholars disagree on the definition. Neil Rhodes emphasized the controversial plot and a play's refusal to take sides on social disputes. A.G. Harmon focused on the tension between law and natural human behavior, while Ernest Schanzer restricted the category to plays presenting a pivotal ethical dilemma that divides the audience equally.
Troilus and Cressida defies simple classification: it was filed among the tragedies in Shakespeare's First Folio but was not listed in the Folio's own table of contents. Unlike the other plays in Boas's original group, it offers no resolution.