lachen
to laugh
To express joy, amusement, or being entertained through facial expressions and sounds.
Der Witz war so gut, dass alle laut lachen mussten.
The joke was so good that everyone had to laugh out loud.

to laugh
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to laugh
To express joy, amusement, or being entertained through facial expressions and sounds.
Der Witz war so gut, dass alle laut lachen mussten.
The joke was so good that everyone had to laugh out loud.

The verb 'lachen' (to laugh) is very often used with the preposition 'über' (about/at) and the accusative. This can mean that you find something funny or that you are making fun of someone.
Wir haben sehr über seinen Witz gelacht.
We laughed a lot at his joke.
To describe the type of laughter, 'lachen' (to laugh) is often combined with adverbs like 'laut' (loudly), 'herzlich' (heartily), or 'leise' (quietly). Colloquially, you also hear the expression 'sich kaputt lachen' (to laugh one's head off).
Sie hat so herzlich gelacht, dass alle mitlachen mussten.
She laughed so heartily that everyone had to laugh along.
When 'lachen über' (to laugh at) refers to a person or their mistakes, it often has a negative, mocking meaning. It then means to make fun of someone.
Die anderen Kinder haben über seine alte Jacke gelacht.
The other children laughed at his old jacket.
The idiom 'nichts zu lachen haben' (to have nothing to laugh about) is a fixed unit. It describes that a person is in a very serious or difficult life situation.
Mit drei kleinen Kindern und wenig Geld hat die Familie gerade nichts zu lachen.
With three small children and little money, the family currently has nothing to laugh about.
to smile
'lachen' and 'lächeln' both show joy or amusement in the face. The most important difference is intensity: with 'lachen' (to laugh), you usually hear a sound; with 'lächeln' (to smile), you only see a friendly or slight facial expression. 'lachen' fits with a joke or a very funny situation. 'lächeln' often fits with greetings, photos, or quiet friendliness. The words can also be confused with 'über': 'über jemanden lachen' (to laugh at someone) can be mean, while 'über etwas lächeln' (to smile at something) often means that you do not take something entirely seriously.

Nach dem Witz musste die ganze Gruppe laut lachen.
After the joke, the whole group had to laugh loudly.
Die Ärztin lächelt freundlich, als sie das Zimmer betritt.
The doctor smiles warmly as she enters the room.