willig
willing
Ready to do something without resistance or hesitation.
Er hat willig zugestimmt, bei den Vorbereitungen zu helfen.
He willingly agreed to help with the preparations.

willing / compliant
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willing
Ready to do something without resistance or hesitation.
Er hat willig zugestimmt, bei den Vorbereitungen zu helfen.
He willingly agreed to help with the preparations.

The adjective 'willig' (willing/compliant) is closely related to the noun 'der Wille' (the will) and the verb 'wollen' (to want). It describes a person who has the will to do something and therefore acts willingly.
Der Student war ein williger Teilnehmer an der Diskussion.
The student was a willing participant in the discussion.
In the meaning of 'bereit sein' (to be willing/ready), 'willig' (willing) is often used in the construction 'willig sein, etwas zu tun' (to be willing to do something). This is followed by a subordinate clause with 'zu' and the infinitive.
Sie war willig, die zusätzliche Arbeit ohne Bezahlung zu übernehmen.
She was willing to take on the additional work without pay.
In the meaning of 'gefügig' (compliant), 'willig' (compliant) is often used in the fixed expression 'ein williges Opfer' (a willing victim). This describes a person who can easily be persuaded or manipulated into something.
Er wurde ein williges Opfer ihrer Pläne.
He became a willing victim of her plans.
ready
'bereit' and 'willig' can both express that someone wants to do something or shows no resistance. 'bereit' sounds neutral and can also mean that someone is prepared. 'willig' puts more emphasis on someone going along with something without resistance. Depending on the context, however, it can also sound somewhat negative, as if someone gives in easily or is too easily influenced. The words are easy to confuse because both have to do with wanting. In normal conversations, 'bereit' is often safer and more natural.

Nach kurzer Diskussion zeigte sich der Kunde willig, den Vertrag zu ändern.
After a short discussion, the customer showed himself willing to change the contract.
Mein Freund ist bereit, morgen früher zum Bahnhof zu fahren.
My friend is willing to go to the train station earlier tomorrow.