aufmerksam
attentive
To be fully concentrated and interested in something; to listen or watch carefully.
Die Kinder hörten der Geschichte aufmerksam zu.
The children listened attentively to the story.

attentive / considerate
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attentive
To be fully concentrated and interested in something; to listen or watch carefully.
Die Kinder hörten der Geschichte aufmerksam zu.
The children listened attentively to the story.

The word is made up of the prefix 'auf', the stem of 'merken' (to notice/remember), and the ending '-sam'. Literally, it therefore means being inclined to notice something or remember something.
Ein aufmerksamer Schüler lernt schneller.
An attentive student learns faster.
The meaning depends strongly on context. If 'aufmerksam' refers to a task or perception, it means 'konzentriert' (concentrated). If it refers to behavior toward people, it means 'rücksichtsvoll' (considerate).
Der Detektiv beobachtete die Szene aufmerksam.
The detective observed the scene attentively.
In the meaning 'konzentriert' (concentrated), 'aufmerksam' is often used as an adverb with verbs of perception. Typical combinations are 'aufmerksam zuhören' (to listen attentively), 'aufmerksam zusehen' (to watch attentively), or 'aufmerksam lesen' (to read attentively).
Bitte lesen Sie den Vertrag aufmerksam durch.
Please read the contract carefully.
To describe a considerate action, the structure 'Es ist aufmerksam von [person in the dative]' is often used. This praises the person who carried out the action.
Es war sehr aufmerksam von Ihnen, an meinen Geburtstag zu denken.
It was very considerate of you to remember my birthday.
The expression 'jemanden auf etwas aufmerksam machen' (to draw someone's attention to something) has a fixed structure. The thing being pointed out is always in the accusative.
Ein Passant machte mich auf den platten Reifen aufmerksam.
A passerby drew my attention to the flat tire.
awake / alert
'aufmerksam' and 'wach' can both describe someone as mentally engaged. 'aufmerksam' says more specifically that someone is listening, watching, or paying attention to details with concentration. In this context, 'wach' means that someone does not seem tired and understands or reacts quickly. You can be 'wach' but still not listen attentively. So 'aufmerksam' fits better for concentration on a specific thing, while 'wach' is more about someone's general mental state.

Heute hört sie im Unterricht besonders aufmerksam zu.
Today she is listening especially attentively in class.
Nach dem Spaziergang bin ich wieder wach und kann weiterarbeiten.
After the walk, I am alert again and can keep working.