färben
to dye
To change the color of something using a dye or color.
Sie hat sich die Haare rot gefärbt.
She dyed her hair red.

to dye / color
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to dye
To change the color of something using a dye or color.
Sie hat sich die Haare rot gefärbt.
She dyed her hair red.

The verb 'färben' (to dye/color) belongs to the word family of 'die Farbe' (color). It literally means to give color to something.
Zu Ostern färben wir immer die Eier.
We always dye eggs at Easter.
When a person actively dyes something, the verb is used transitively: 'jemand färbt etwas' (someone dyes something). When something changes its color by itself, it is used reflexively: 'etwas färbt sich' (something changes color).
Ich färbe mein altes T-Shirt schwarz.
I am dyeing my old T-shirt black.
When you dye something on your own body, a dative pronoun is often added to make the connection to the person clear. The most common form is 'sich (Dat.) die Haare (Acc.) färben' (to dye one's hair).
Ich habe mir die Haare blond gefärbt.
I dyed my hair blond.
When something changes color naturally, the subjects are often parts of nature. Typical examples are 'die Blätter' (the leaves), 'der Himmel' (the sky), or 'die Wolken' (the clouds).
Die Blätter an den Bäumen haben sich schon gelb gefärbt.
The leaves on the trees have already turned yellow.
This reflexive meaning refers almost exclusively to a person's face or cheeks turning red from embarrassment, shame, or joy.
Sein Gesicht färbte sich rot, als alle über seinen Witz lachten.
His face turned red when everyone laughed at his joke.
to sound / to tint / to boast
'färben' and 'tönen' can both mean that hair gets a different color. 'färben' is more general and often stronger or more permanent. 'tönen' is used especially for hair when the color is lighter and washes out again after some time. The words are easy to confuse because both have to do with hair color at the hairdresser's or at home. If the new color is meant to be permanent, 'färben' usually fits. If it is only temporary, 'tönen' is more precise.

Vor dem Urlaub hat sich Lena die Haare dunkel gefärbt.
Before the vacation, Lena dyed her hair dark.
Für die Feier lässt sich Max die Haare nur leicht tönen.
For the party, Max is only having his hair lightly tinted.