
lächeln
to smile
To move the mouth into an expression of joy, friendliness, or amusement without making a loud sound.
Als sie ihn sah, musste sie sofort lächeln.
When she saw him, she immediately had to smile.
to smile
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to smile
To move the mouth into an expression of joy, friendliness, or amusement without making a loud sound.
Als sie ihn sah, musste sie sofort lächeln.
When she saw him, she immediately had to smile.

to smile at something
To regard something or someone with slight superiority or indulgence; to not take something entirely seriously.
Der erfahrene Meister konnte über die Fehler des Lehrlings nur milde lächeln.
The experienced master could only smile mildly at the apprentice's mistakes.
The word 'lächeln' (to smile) is a diminutive form of 'lachen' (to laugh). So it describes, in a way, a 'small laugh'—quiet and only with the mouth.
Ein Lachen kann man hören, ein Lächeln nur sehen.
A laugh can be heard; a smile can only be seen.
If you want to express that you are smiling directly at a person, the separable verb 'anlächeln' (to smile at) is often used. The object then takes the accusative.
Der Barista lächelte mich freundlich an.
The barista smiled at me in a friendly way.
'Lächeln' (to smile) is often combined with adverbs that describe the type of smile. Common examples are 'freundlich lächeln' (to smile in a friendly way), 'glücklich lächeln' (to smile happily), or 'verlegen lächeln' (to smile shyly/awkwardly).
Als er das Kompliment hörte, lächelte er verlegen.
When he heard the compliment, he smiled shyly.
In the meaning 'to not take something seriously', 'lächeln' (to smile) is always used with the preposition 'über' (about/at) and the accusative. The person smiling often feels superior.
Sie konnte über seine naiven Vorstellungen nur müde lächeln.
She could only smile wearily at his naive ideas.
to laugh
Lachen and lächeln both express joy, friendliness, or amusement. Lachen normally also involves audible sounds, while lächeln refers to a silent facial expression. Use lachen for an audible reaction and lächeln when the person is reacting only in a friendly or quiet way.

Als der Schauspieler den Witz erzählte, musste sie nur leise lächeln.
When the actor told the joke, she could only smile quietly.
Als der Schauspieler den Witz erzählte, begannen alle laut zu lachen.
When the actor told the joke, everyone started laughing loudly.