untergehen
to set
Referring to celestial bodies such as the sun or moon: to disappear at the horizon.
Wir saßen am Strand und sahen zu, wie die Sonne im Meer unterging.
We sat on the beach and watched as the sun set into the sea.

to set / sink / perish
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to set
Referring to celestial bodies such as the sun or moon: to disappear at the horizon.
Wir saßen am Strand und sahen zu, wie die Sonne im Meer unterging.
We sat on the beach and watched as the sun set into the sea.

'Untergehen' (to set/sink/perish) is a separable verb. In a main clause, the prefix 'unter' goes at the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Die Sonne geht heute besonders schön unter.
The sun is setting especially beautifully today.
The verb is made up of 'unter' (under/down) and 'gehen' (to go). The literal meaning 'nach unten gehen' (to go downward) helps you understand most meanings: the sun goes below the horizon, a ship goes under water.
Wir sahen zu, wie die Sonne langsam am Horizont unterging.
We watched as the sun slowly set on the horizon.
The perfect tense of 'untergehen' (to set/sink/perish) is always formed with the auxiliary verb 'sein' (to be), because it describes a change of state or movement. So it is 'ist untergegangen' (has set/sunk/perished).
Das Römische Reich ist vor langer Zeit untergegangen.
The Roman Empire perished a long time ago.
The meaning of 'untergehen' (to set/sink/perish) can often be recognized from the subject. 'Die Sonne' (the sun) sets (disappears at the horizon), 'ein Schiff' (a ship) sinks, and 'ein Reich' (an empire) perishes (is destroyed).
Viele alte Zivilisationen sind untergegangen.
Many ancient civilizations have perished.
In the figurative meaning 'nicht bemerkt werden' (to go unnoticed), 'untergehen' (to get lost) is often used with the preposition 'in' + dative. One gets lost 'in der Menge' (in the crowd), 'im Lärm' (in the noise), or 'in der Masse' (in the masses).
Seine leise Bemerkung ging im allgemeinen Gelächter unter.
His quiet remark got lost in the general laughter.
to disappear
'verschwinden' and 'untergehen' can both express that something is no longer visible or no longer exists. 'verschwinden' is neutral and very general: a person, an object, or a feeling can disappear. 'untergehen' often has a more specific image: the sun sets, a ship sinks in the water, or an empire perishes. It can also mean that something goes unnoticed in a crowd or mass. The words can be confused because both describe something being gone. However, 'untergehen' only fits in certain situations; 'verschwinden' is the more general choice.

Am Strand bleiben wir sitzen, bis die Sonne untergeht.
We stay sitting on the beach until the sun sets.
Nach dem Gespräch verschwindet die Chefin schnell in ihr Büro.
After the conversation, the boss quickly disappears into her office.