Why go
The phrase “city of peace and justice” has a history you can trace. The early peace conferences led to the Permanent Court of Arbitration; the Peace Palace gave that institution a home; and later courts extended The Hague's role. Knowing this sequence makes the institutions easier to distinguish when you encounter their names.
The full story
The first Hague Peace Conference met in 1899, followed by another in 1907. The meetings advanced formal ways for states to address disputes and helped establish the Permanent Court of Arbitration. Andrew Carnegie funded the Peace Palace as its home. The palace later also became the seat of the Permanent Court of International Justice, whose successor after the Second World War was the United Nations' International Court of Justice. The International Criminal Court represents another, separate chapter in the city's legal role. These institutions do different work: the International Court of Justice deals with disputes submitted by states and gives advisory opinions, while the criminal court concerns individual criminal responsibility. Read the city through these distinctions, from the Peace Palace outward, rather than assuming every international organisation is part of one court. The palace is a physical anchor for this history, but the names describe separate institutions and remits. Check public access for each site before planning an interior visit.