das Armband
bracelet
A piece of jewelry worn around the wrist.
Sie trägt ein schönes silbernes Armband.
She is wearing a beautiful silver bracelet.

bracelet / wristband
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bracelet
A piece of jewelry worn around the wrist.
Sie trägt ein schönes silbernes Armband.
She is wearing a beautiful silver bracelet.

The word 'Armband' (bracelet / wristband) is a simple compound of 'der Arm' (arm) and 'das Band' (band). So it is literally a band that you wear on your arm.
Zum Geburtstag habe ich ein neues Armband bekommen.
I got a new bracelet for my birthday.
The most common verbs used with 'Armband' (bracelet / wristband) are 'tragen' (to wear), 'anlegen' (to put on), or 'ummachen' (to put on / fasten), and 'abnehmen' (to take off).
Er hat sein Armband vor dem Schwimmen abgenommen.
He took off his bracelet before swimming.
Context tells you whether a piece of jewelry or a functional band is meant. Words like 'golden' (gold), 'Perlen' (pearls), or 'Schmuck' (jewelry) point to meaning 1. Words like 'Eintritt' (entry), 'Krankenhaus' (hospital), or 'Festival' (festival) point to meaning 2.
Jeder Patient erhält bei der Aufnahme ein Armband mit seinem Namen.
Every patient receives a wristband with their name upon admission.
wristband / ribbon
'das Armband' and 'das Bändchen' can both mean something worn on the wrist. 'Armband' often sounds like jewelry, but it can also be a band for entry or checking access. 'Bändchen' sounds smaller and simpler, for example a fabric or plastic wristband at a concert or festival. In addition, a 'Bändchen' can also be a small narrow ribbon in general, not only something worn on the wrist. The words can be confused especially at events. For jewelry, 'Armband' is more natural; for a simple entry wristband, it is often 'Bändchen'.

Zum Geburtstag schenkte Lena ihrer Schwester ein silbernes Armband.
For her birthday, Lena gave her sister a silver bracelet.
Am Eingang bekam Nele ein grünes Bändchen für das Konzert.
At the entrance, Nele got a green wristband for the concert.