What is the circumstance (ḥāl)?

A verb tells you what happened. The حَال tells you the condition of the one it happened to, or by, at that moment.

جَاءَ زَيْدٌ رَاكِبًا (jā'a zaydun rākiban) — Zayd came riding. رَاكِبًا is not who came and not what he came on; it is the state he was in while coming.

A ḥāl may be one word, a whole sentence, or a preposition with its noun.

The word being described is the صَاحِب الْحَال (ṣāḥib al-ḥāl), the owner of the state, and it is normally definite while the ḥāl itself is indefinite.

A ḥāl is not always one word. It comes in three shapes: a single word — رَاكِبًا; a whole sentence — جَاءَ زَيْدٌ يَضْحَكُ, Zayd came laughing; or a quasi-sentence, a preposition with its noun.

Do not confuse the حَال with the نَعْت (the adjective).

Both describe. The difference is what they attach to and how long the description lasts.

A نَعْت describes a noun as a standing quality, and copies it in everything — definiteness included: جَاءَ الرَّجُلُ الطَّوِيلُ, the tall man came. He is tall whether or not he is coming.

A حَال describes a passing state tied to the moment of the action, is indefinite while its owner is definite, and is always manṣūb: جَاءَ الرَّجُلُ رَاكِبًا. He was riding only as he came.