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.

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
'unterhalten' and 'amüsieren' can both mean that someone makes others have fun or keeps them pleasantly occupied. 'unterhalten' is a bit more general: a film, a conversation, or a program can be interesting and make the time enjoyable. 'amüsieren' puts more emphasis on fun, laughter, or being in a good mood. Learners may confuse the words because 'jemanden unterhalten' often seems almost like 'jemanden amüsieren'. If it is only about pleasant occupation, 'unterhalten' fits well; if something is truly funny, 'amüsieren' is more precise.

Die kleinen Geschichten des Clowns amüsieren die Kinder sehr.
The clown's little stories amuse the children a lot.
Der Musiker unterhält die Gäste während des Abendessens.
The musician entertains the guests during dinner.