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.

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 take
'nehmen' and 'fassen' can both mean grasping something or someone with your hand. The difference lies in the focus: 'nehmen' usually means taking something toward yourself or getting hold of it. 'fassen' puts stronger emphasis on gripping and holding on. That is why you are more likely to say 'ein Glas nehmen' (to take a glass) when you want to use it, but 'jemanden am Arm fassen' (to take someone by the arm) when you put your hand on their arm and hold it. With people, it can also be about arrest or capture: 'gefangen nehmen' (to take prisoner) is possible; 'einen Täter fassen' means that you catch and arrest the offender. The words can be confused because both have to do with grasping, but 'nehmen' is more general and 'fassen' sounds more precise when it comes to holding on.

Sie fasste das Kind vorsichtig an der Hand.
She carefully took the child by the hand.
Beim Einkaufen nehme ich heute die kleineren Tomaten.
When shopping today, I'm taking the smaller tomatoes.