Unbounded describes something that has no edge, no ceiling, no point where it stops. In real use it is almost always about feelings and qualities rather than physical space: unbounded joy, unbounded enthusiasm, unbounded ambition, unbounded generosity. The picture behind it is a field with no fence — a bound is a border, and this thing has none. It is a formal, written word. You will find it in books, speeches and serious articles; you will not hear it from a friend telling you about their weekend, who would say 'so happy' or 'crazy about it'. It is a warm word more often than not: unbounded usually praises. Compare boundless, which means almost the same thing and is a little more common; and note that when the same idea turns critical, English tends to switch to unbridled or unchecked instead. In mathematics unbounded has a strict technical sense — a set or function with no finite limit — but your students will meet the everyday version first.