gefallen
to please
To be liked by someone or to be perceived as pleasant. The person who likes something is in the dative.
Mir gefällt deine neue Frisur sehr gut.
Your new hairstyle really pleases me.

to please / to fall
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to please
To be liked by someone or to be perceived as pleasant. The person who likes something is in the dative.
Mir gefällt deine neue Frisur sehr gut.
Your new hairstyle really pleases me.

The verb 'gefallen' (to please, to fall) is inseparable. The prefix 'ge-' remains firmly attached to the verb stem 'fallen' in all conjugated forms and sentence positions.
Er gefiel sich schon immer in der Rolle des Anführers.
He had always reveled in the role of the leader.
In the most common meaning, the person who likes something is in the dative, and the thing that is liked is in the nominative. So the sentence structure is the reverse of the English formulation 'I like it'.
Das neue Album der Band gefällt mir sehr.
The band's new album really pleases me.
In the meaning 'to die in battle', 'gefallen' (to fall in battle) is used almost only in military or historical contexts. It is a formal and euphemistic paraphrase for 'sterben' (to die).
Sein Großvater ist im Zweiten Weltkrieg gefallen.
His grandfather fell in the Second World War.
The idiom 'sich (Dativ) etwas (Akkusativ) gefallen lassen' (to put up with something) means to tolerate negative treatment. Pay attention to the fixed structure with the reflexive pronoun in the dative and the verb 'lassen' (to let).
Lass dir sein unhöfliches Verhalten nicht gefallen!
Don't put up with his rude behavior!
The expression 'sich in etwas (Dativ) gefallen' (to revel in something) describes that someone visibly enjoys a role or behavior. The preposition 'in' is fixed with the verb here.
Sie gefällt sich in ihrer Rolle als strenge Chefin.
She revels in her role as the strict boss.
to fall
'fallen' and 'gefallen' look very similar, but most of the time they mean something different. 'fallen' usually means that something goes downward or that someone falls. In texts about war, 'fallen' can also mean that someone is killed in battle. 'gefallen' can mean almost the same thing in this war context, but it sounds more formal. Much more often, however, 'gefallen' means that something is beautiful or pleasant to someone. Then the person is in the dative, for example 'mir gefällt' (I like it / it pleases me). The risk of confusion is high because 'gefallen' also looks like a form of 'fallen'. So pay attention to the context and to the dative.

Das neue Kleid gefällt meiner Schwester sehr.
My sister really likes the new dress.
Viele junge Männer fielen in diesem Krieg.
Many young men fell in this war.