gebunden
bound
Physically fastened or tied together with a rope, string, or something similar.
Der Blumenstrauß war mit einem schönen Band gebunden.
The bouquet was bound with a beautiful ribbon.

bound
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bound
Physically fastened or tied together with a rope, string, or something similar.
Der Blumenstrauß war mit einem schönen Band gebunden.
The bouquet was bound with a beautiful ribbon.

'Gebunden' (bound) is the past participle of the verb 'binden' (to bind/tie). All meanings of the adjective derive from the basic idea that something is fastened, connected, or limited in its freedom.
Er hat die alten Zeitungen zu einem Paket gebunden.
He tied the old newspapers into a bundle.
When 'gebunden' (bound) expresses an obligation, it is always used with the preposition 'an' and the accusative. The structure is: 'an etwas gebunden sein' (to be bound to something).
Als Mitarbeiter sind wir an die Regeln des Unternehmens gebunden.
As employees, we are bound to the company’s rules.
In a chemical context, the preposition 'an' is also used. The difference from obligation is made clear by the nouns: here, atoms or molecules are 'an' (to) other atoms bonded.
Im Wassermolekül (H₂O) ist ein Sauerstoffatom an zwei Wasserstoffatome gebunden.
In the water molecule (H₂O), one oxygen atom is bonded to two hydrogen atoms.
In connection with books, 'gebundene Ausgabe' (hardcover edition) is a fixed term for a book with a hard cover. It is the opposite of 'broschierte Ausgabe' (paperback edition).
Viele Sammler bevorzugen die gebundene Erstausgabe eines Romans.
Many collectors prefer the hardcover first edition of a novel.
When 'gebunden' (thickened) is used in cooking, it means that a liquid such as a soup or sauce has been made thicker. The verb for this is 'binden' (to thicken).
Für eine cremig gebundene Suppe kann man etwas Sahne oder Mehl verwenden.
For a creamy, thickened soup, you can use a little cream or flour.
In finance, 'gebunden' (tied up) means that money is not immediately available. Here, the preposition 'in' is often used: capital is 'in' (in) an investment tied up.
Sein Geld ist langfristig in Aktien gebunden.
His money is tied up in stocks for the long term.
free
'frei' and 'gebunden' can be opposites when it comes to movement or decisions. 'frei' then means that someone is not being held or can decide for themselves. 'gebunden' means either physically tied or obligated to a rule, a contract, or a promise. So the contrast depends on the context: a person can be physically free, but professionally bound to a contract. 'frei' is broader, while 'gebunden' names the type of restriction more precisely.

Wegen des Vertrags ist mein Bruder noch ein Jahr an die Firma gebunden.
Because of the contract, my brother is bound to the company for another year.
Am Wochenende bin ich frei und kann selbst entscheiden, was ich mache.
I’m free on the weekend and can decide for myself what I’ll do.