
sich amüsieren
to enjoy oneself
To have a good time, have fun, and be entertained in a pleasant way.
Wir haben uns auf der Party köstlich amüsiert.
We had a wonderfully good time at the party.
to amuse / enjoy oneself
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to enjoy oneself
To have a good time, have fun, and be entertained in a pleasant way.
Wir haben uns auf der Party köstlich amüsiert.
We had a wonderfully good time at the party.

to amuse
To entertain someone or make them laugh.
Der Komiker verstand es, sein Publikum zu amüsieren.
The comedian knew how to amuse his audience.

to make fun of
To make fun of someone or something, often in a slightly condescending or mocking way.
Anstatt ihm zu helfen, amüsierten sich die anderen über sein Missgeschick.
Instead of helping him, the others made fun of his mishap.
The structure of the sentence completely changes the meaning. 'Sich amüsieren' means to have fun. 'Jemanden amüsieren' means to entertain someone. 'Sich über jemanden amüsieren' means to make fun of a person.
Die Kinder amüsierten sich im Park.
The children enjoyed themselves in the park.
Do not confuse 'sich amüsieren' (to have fun/enjoy oneself) with 'sich über etwas/jemanden amüsieren' (to make fun of something/someone). The second expression can be negative or hurtful. The preposition 'über' (about/over) is crucial here.
Wir haben uns bei dem Fest gut amüsiert.
We had a good time at the festival.
When you want to express that you had a lot of fun, 'sich amüsieren' (to enjoy oneself) is often intensified with adverbs such as 'köstlich' (wonderfully), 'prächtig' (splendidly), or 'königlich' (royally).
Ich habe mich bei dem Theaterstück köstlich amüsiert.
I had a wonderfully good time at the play.
The expression 'sich über jemanden amüsieren' (to make fun of someone) often has a slightly negative or condescending undertone. It describes mockery rather than friendly laughter.
Anstatt ihm aufzuhelfen, amüsierten sich die Zuschauer nur über seinen Sturz.
Instead of helping him up, the spectators only made fun of his fall.
to converse / entertain / maintain
Both words can mean providing others with an enjoyable time. Unterhalten is the broader and more neutral choice, including music, conversations, or a program; amüsieren places greater emphasis on fun and cheerfulness, often involving funny moments. Choose amüsieren when the comic effect is the main focus.

Der Gastgeber amüsierte die Gäste mit lustigen Geschichten.
The host amused the guests with funny stories.
Der Gastgeber unterhielt die Gäste mit Musik und Gesprächen.
The host entertained the guests with music and conversation.