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olvidar algo: to forget something:
Olvidé cerrar la puerta. -
olvidarse algo (dativo ético): to forget [to take/bring] something:
Me olvidé las llaves(I forgot the keys [≈I left them by recklessness]). -
olvidarse de algo: to forget (about) something (it is used synonymously with olvidar most of the time):
Me olvidé de que hoy era tu cumpleaños. -
olvidársele: pretty much the same as the 1st or the 3rd. Because you’re not the subject, you’re kind of distancing yourself from the responsibility (similarly to “It slipped my mind”):
Se me olvidó que me habías pedido que te mandara el mensaje.
Take a look at the definitions in Diccionario del estudiante.