einschlafen
to fall asleep
To enter the state of sleep.
Das Baby ist nach wenigen Minuten auf meinem Arm eingeschlafen.
The baby fell asleep in my arms after just a few minutes.

to fall asleep / go numb
Loading your saved state.
to fall asleep
To enter the state of sleep.
Das Baby ist nach wenigen Minuten auf meinem Arm eingeschlafen.
The baby fell asleep in my arms after just a few minutes.

'Einschlafen' (to fall asleep) is a separable verb. The prefix 'ein-' goes to the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Ich lese jeden Abend, bis ich einschlafe.
I read every evening until I fall asleep.
'Einschlafen' (to fall asleep) describes a transition into another state. That is why the perfect tense is always formed with the auxiliary verb 'sein' (to be), not with 'haben' (to have).
Bist du gestern gut eingeschlafen?
Did you fall asleep easily yesterday?
When a body part 'einschläft' (goes numb), the person experiencing it is in the dative. The subject of the sentence is the body part.
Mir ist beim langen Sitzen der Fuß eingeschlafen.
My foot went numb while I was sitting for a long time.
In this figurative meaning, the subject is usually something abstract that slowly stops. Typical subjects are 'der Kontakt' (the contact), 'die Freundschaft' (the friendship), 'das Gespräch' (the conversation), or 'die Begeisterung' (the enthusiasm).
Nachdem er wegzog, ist unsere Freundschaft langsam eingeschlafen.
After he moved away, our friendship slowly petered out.
In the meaning of 'sterben' (to die), 'einschlafen' (to pass away) is a very gentle and respectful euphemism. It is used to express that someone died peacefully.
Er ist nach langer Krankheit im Kreise seiner Familie eingeschlafen.
After a long illness, he passed away surrounded by his family.
to sleep
'schlafen' and 'einschlafen' are easy to confuse because both have to do with sleep. 'schlafen' describes the state: you are already asleep. 'einschlafen' describes the transition: you get tired and start to sleep. So you say: 'Ich schlafe seit zwei Stunden' (I have been sleeping for two hours), but 'Ich bin um zehn Uhr eingeschlafen' (I fell asleep at ten o'clock). Both words can also be used for body parts: a foot 'schläft' or 'ist eingeschlafen' when it goes numb and tingles. In this context, 'eingeschlafen' often sounds more natural when the state has just come about.

Im Zug schläft sie oft schon nach wenigen Minuten ein.
On the train, she often falls asleep after just a few minutes.
Zu Hause haben wir nach der Nachtschicht lange geschlafen.
At home, we slept for a long time after the night shift.