The judgment (卦辭, guàcí, "words of the hexagram") is the short text the ancient book places beside each of the 64 hexagrams. It's the voice of the whole hexagram: it sums up the situation and often says, in very few words, whether the moment is favorable and what to do.
What it sounds like
It's terse. The judgment of Activity is just four characters, 元亨利貞, roughly "Great success. Constancy is favorable" (see four virtues). Others are longer and carry concrete images: a journey, a sacrifice, crossing the great river. Many use fixed omen formulas, like those of good fortune and misfortune.
In your reading
You always read the judgment of your primary hexagram. With no moving lines, it's nearly your whole answer, together with the image. With moving lines, the judgment sets the frame and the line text of each moving line gives the detail.
In Cambios, the poetic layer of each hexagram is written from this Chinese text, and the practical layer tells you what it means for your question.
Origins: tradition and scholarship
Tradition says King Wen wrote the judgments while he was held prisoner. Scholars see these texts as part of the oldest core of the book, the Zhouyi, compiled over the Western Zhou period, with no single proven author. The commentary that explains them is the Tuan.