merken
to notice
To perceive, notice, or realize something; to become aware of something.
Ich habe erst zu spät gemerkt, dass ich meinen Schlüssel vergessen habe.
I only noticed too late that I had forgotten my key.

to notice / remember
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to notice
To perceive, notice, or realize something; to become aware of something.
Ich habe erst zu spät gemerkt, dass ich meinen Schlüssel vergessen habe.
I only noticed too late that I had forgotten my key.

This verb changes its meaning depending on whether it is reflexive or not. 'Etwas merken' (to notice something) means to perceive something. 'Sich etwas merken' (to remember something) means to memorize something or keep it in memory.
Ich merke, dass es kälter wird.
I notice that it's getting colder.
The structure for 'sich etwas einprägen' (to memorize something) is always: `sich (dative) + etwas (accusative) + merken`. The person who remembers something is in the dative, and the information is in the accusative.
Merk dir bitte meine neue Adresse.
Please remember my new address.
In the meaning 'wahrnehmen' (to perceive), 'merken' (to notice) is very often used with a subordinate clause that begins with 'dass' (that). This construction expresses that someone realizes a situation or fact.
Er hat gar nicht gemerkt, dass alle Leute ihn ansahen.
He didn't notice at all that everyone was looking at him.
The expression 'sich etwas anmerken lassen' (to let something show) is a fixed expression. It is usually used in the negative ('sich nichts anmerken lassen' — to not let anything show) to express that someone successfully hides their feelings.
Sie war sehr aufgeregt, aber sie hat sich nichts anmerken lassen.
She was very excited, but she didn't let anything show.
to show
'zeigen' and 'merken' overlap when feelings or thoughts become visible through behavior. 'zeigen' describes more actively or neutrally that one lets a feeling be seen: a face shows joy, someone shows anger. With 'sich etwas anmerken lassen' (to let something show), the point is whether others can recognize something in a person. This form is often used in the negative: 'Man lässt sich nichts anmerken' (you don't let anything show). So 'zeigen' fits more directly for the visible feeling, while 'sich anmerken lassen' emphasizes whether you reveal that feeling outwardly.

Die Gäste merken nicht, dass Lisa wegen der Rede nervös ist.
The guests don't notice that Lisa is nervous about the speech.
Sein Lächeln zeigt, dass er sich über die Nachricht freut.
His smile shows that he is happy about the news.