Caer vs. caerse

In Spanish, the verbs 'caer' and 'caerse' both involve the action of falling, but they are used in slightly different contexts and ways.

Caer

'Caer' means 'to fall' or 'to drop' and generally refers to objects or subjects falling due to external factors.
El vaso cayó de la mesa.
(The glass fell off the table.)
La manzana cayó del árbol.
(The apple fell from the tree.)
La lluvia comenzó a caer lentamente.
(The rain started to fall slowly.)

Caerse

'Caerse' is a reflexive form of 'caer', and it is typically used when referring to people or animals falling, often implying that they lost their balance or footing by themselves.
Me caí por las escaleras.
(I fell down the stairs.)
El niño se cayó mientras corría.
(The child fell while running.)
Mi abuela se cayó en la calle ayer.
(My grandmother fell in the street yesterday.)

Summary

In summary, while both 'caer' and 'caerse' refer to the action of falling, 'caer' is generally used for objects or external causes of falling, and 'caerse' is used reflexively to indicate that someone or something fell by themselves.