Quick answer
dejar = to leave something somewhere, to lend, or to let someone do something.
dejar de + infinitive = to stop doing something.
With an infinitive and no de, you are giving permission. With de, you are quitting.
Side by side
Déjame hablar. / Deja de hablar.Let me speak versus stop talking — opposite meanings, one preposition apart.
Dejé el trabajo. / Dejé de trabajar.I quit the job versus I stopped working altogether.
No dejes de llamarme.Don't fail to call me — a standard way of insisting, not a negative order.
Memory trick
See an infinitive? With <i>de</i> it means stop, without <i>de</i> it means let. No infinitive at all means leave or lend.
Typical mistakes
Dejé fumar hace un año.
Dejé de fumar hace un año.
Quitting needs de. Without it the sentence reads as "I let someone smoke".
Deja de mí terminar.
Déjame terminar.
Permission takes a direct object pronoun and no de: déjame, déjanos, déjalo.
Check yourself
Fill in: "Por favor, ___ interrumpirme."