
fassen
to grasp
To take hold of and hold onto someone or something with the hand or hands.
Er fasste meine Hand, um mich über die Straße zu führen.
He grasped my hand to lead me across the street.
to grasp / catch
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to grasp
To take hold of and hold onto someone or something with the hand or hands.
Er fasste meine Hand, um mich über die Straße zu führen.
He grasped my hand to lead me across the street.

to catch
To catch and arrest a criminal or fugitive.
Die Polizei konnte den Dieb nach kurzer Verfolgung fassen.
The police were able to catch the thief after a short chase.

to grasp
To mentally grasp, understand, or emotionally process something. Often used in the negative form.
Ich kann es immer noch nicht fassen, dass ich im Lotto gewonnen habe.
I still can't grasp that I won the lottery.

to compose oneself
To regain self-control and calm after a shock or excitement.
Nach der Schreckensnachricht musste sie sich erst einmal fassen, bevor sie sprechen konnte.
After the terrible news, she first had to compose herself before she could speak.

to hold
To be able to contain a certain volume or amount of contents.
Dieses Fass fasst 200 Liter Wein.
This barrel holds 200 liters of wine.

to make a decision
To make a decision, often after careful consideration.
Nach langem Nachdenken fasste sie den Entschluss, ihren Job zu kündigen.
After thinking for a long time, she made the decision to quit her job.

to put into words
To put thoughts, feelings, or an idea into words; to formulate something.
Er versuchte, seine Dankbarkeit in Worte zu fassen.
He tried to put his gratitude into words.
Many meanings of 'fassen' (to grasp / catch) share the basic idea of enclosing or being able to contain. You can grasp a hand (enclose it), a thought is grasped by the mind (understood), and you compose yourself again (bring your feelings under control).
Sie konnte die Nachricht kaum fassen.
She could hardly grasp the news.
When describing where you touch or grasp someone or something, the preposition 'an' or 'bei' is used with the dative. This is a fixed structure.
Die Mutter fasste das Kind an der Hand.
The mother took the child by the hand.
In the meaning of 'fangen' (to catch), 'fassen' (to catch) is used almost exclusively for criminals, suspects, or escaped animals. It is typical language in police reports and news.
Die Polizei hofft, den Einbrecher bald zu fassen.
The police hope to catch the burglar soon.
To express amazement or disbelief, 'fassen' (to grasp) is very often used in the negative form with 'können' (can). The phrase 'Ich kann es nicht fassen!' (I can't believe it! / I can't grasp it!) is very common.
Ich kann es nicht fassen, dass du das wirklich getan hast!
I can't grasp that you really did that!
The reflexive form 'sich fassen' (to compose oneself) means to regain self-control after a shock or strong emotion. The person calming down is the subject of the sentence.
Atme tief durch und versuch, dich zu fassen.
Take a deep breath and try to compose yourself.
The expression 'einen Entschluss fassen' (to make a decision) is a fixed collocation and a more formal way of saying 'sich entscheiden' (to decide). Other similar expressions are 'einen Beschluss fassen' (to pass a resolution) or 'einen Plan fassen' (to make a plan).
Der Stadtrat hat den Beschluss gefasst, eine neue Schule zu bauen.
The city council has made the decision to build a new school.
The phrase 'etwas in Worte fassen' (to put something into words) means to express thoughts or feelings. It is often used when this is especially difficult.
Er konnte seine Freude kaum in Worte fassen.
He could hardly put his joy into words.
to hold
Both words can describe grasping something with the hand. “Fassen” usually emphasizes the moment when you take hold of something. “Halten,” by contrast, emphasizes that you do not let go afterward.

Sie fasst das Glas vorsichtig mit der rechten Hand.
She carefully grasps the glass with her right hand.
Sie hält das Glas fest in der rechten Hand.
She holds the glass firmly in her right hand.