greifen
to grab
To take, grasp, or seize something with the hand.
Er griff nach dem Buch, das auf dem obersten Regal lag.
He reached for the book that was on the top shelf.

to grab / take effect
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to grab
To take, grasp, or seize something with the hand.
Er griff nach dem Buch, das auf dem obersten Regal lag.
He reached for the book that was on the top shelf.

'Greifen' (to grab) is the base word for many important verbs such as 'begreifen' (to understand), 'angreifen' (to attack), or 'eingreifen' (to intervene). The basic idea of 'grasping' or 'comprehending' is often recognizable.
Es ist schwer zu begreifen, was passiert ist.
It is hard to understand what happened.
To emphasize movement toward an object, the construction 'greifen nach' + dative is often used. If you take the object directly, an accusative object can also follow.
Das Kind griff nach der Hand seiner Mutter.
The child reached for his mother's hand.
The expression 'zu etwas greifen' (to resort to something) means to choose a certain means or measure, often out of necessity. The noun is in the dative.
Bei starken Schmerzen sollte man zu einem Medikament greifen.
For severe pain, one should resort to medication.
In this meaning, 'greifen' (to take effect) is used impersonally, and the subject is always an abstract concept such as a law, rule, or reform. It means 'wirksam werden' (to become effective) or 'gelten' (to apply).
Die neue Regelung greift ab dem ersten Januar.
The new regulation takes effect on the first of January.
The expression 'um sich greifen' (to spread) is a fixed expression and means that something spreads quickly and often uncontrollably. It is mostly used for negative things such as fire, diseases, or panic.
Die Angst vor dem Virus griff schnell um sich.
Fear of the virus spread quickly.
to take
'nehmen' and 'greifen' can both have to do with the hand and an object. 'nehmen' describes the result more: afterward, you have the object with you or are using it. 'greifen' emphasizes the movement of the hand and the act of grasping. You can reach for something without having securely taken it yet. The words can be confused because both can occur in the same action. In practical terms: 'greifen' is the hand movement, while 'nehmen' is taking something into your possession.

Im Dunkeln greift er nach dem Schlüssel auf dem Tisch.
In the dark, he reaches for the key on the table.
Vor dem Termin nimmt sie ihre Tablette mit einem Glas Wasser.
Before the appointment, she takes her tablet with a glass of water.