Prophet Moses (p.b.u.h.), his name was Prophet Musa (peace be upon him) in Islam, the son of Imran was born in Egypt 450 years after Prophet Yusuf [Joseph] (p.b.u.h.). In Egypt, at that time the king were known as Fir'awn [Pharaoh].
The followers of the Prophet Yaqub [Jacob] (p.b.u.h.). father of Prophet Yusuf (p.b.u.h.) are called Bani isra'il [Israelites]. Bani Isra'il had lived in Egypt since the days of Prophet Yusuf (p.b.u.h.). Prophet Yaqub [jacob] was known as "Isra'il"
The Prophet Musa (p.b.u.h.) was born during the critical period. His mother managed to conceal him for 3 months and when she could not manage anymore, she was inspired by Allah to put Musa (SAW) into a specially made box and threw it into the river. Holy Qur'an Sura Taha (20:38-39)
"Behold! We sent to thy mother, by inspiration, The message: :'Throw (the child) into the chest, and throw (the chest) into the river: The river will cast him up on the bank, and he will be taken up by one who is an enemy to Me and an enemy to him'; But I cast (the garment of) love over thee from Me: and (this) in order that thou mayest be reared under Mine eye.
Maryam, Musa's (SAW) sister was asked to watch the floating box from a distance to avoid suspicion. Sura Qasas (28:11)
And she said to the sister of [Musa (SAW)], "Follow him." So she (the sister) watched him in the character of a stranger, and they knew not.
The box reached the other shore and one of the members of the Fir'awn's family found it. Musa (SAW) was then taken to Fir'awn's wife and she was very glad to have the baby adopted Sura Qasas (28:8-9)
Then the people of Pharaoh picked him up (from the river): (It was intended) that [Musa (SAW)] should be to them an adversary and a cause of sorrow : For Pharaoh and Haman and (all) their hosts were men of sin.
The Wife of Pharaoh said: "(Here is) a joy of the eye, for me and for thee: Slay him not. It may be that he will be of use to us, or we may adopt him as a son." And they perceived not (what they were doing)!

Maryam went to Fir'awn's falace and suggested a nanny to look after the baby and the woman was Musa (SAW) mother. Sura Qasas (28:12)
And we ordained that he refused suck at first, until (his sister came up and) said: "shall I point out to you the people of a house that will nourish and bring him up for you and be sincerely attached to him?"
When he grew up and while he was walking, he came across an Egyptian who was beating an Israelite Musa (SAW) gave the Egyptian a blow that killed him accidentally. Sura Qasas (28:15)
And he entered the City at a time when its people were not watching: and he found there two men fighting, one of his religion, and the other, of his foes. Now the man of his own religion appealed to him against his foes, and Musa (SAW) struck him with his fist and made an end of him. he said: "This is a work of Evil (satan): for he is an enemy that manifestly misleads!"
He left Fir'awn's house and went to Midian. Sura Qasas (28:22-28)
Then, when he turned his face towards (the land of ) Madyan, he said: "I do hope that my Lord will show me the smooth and straight path."
And when he arrived at the watering (place) in madyan, he found there a group of men watering (their flocks), and besides them he found two women who were keeping back (their flocks), He said: "What is the matter with you?" They said: "We cannot water (our flocks) until the sheperds take back (their flocks) :and our father is a very old man."
So he watered (their flocks) for them; then he turned back to the shade, and said: " O my Lord ! Truly am I in (desperate) need of any good that thou dost send me !"...
Afterwards one of the (damsels) came (back) to him, walking bashfully. She said: "My father invites thee he may reward thee for having watered (our flocks) for us." So when he came to him and narrated the story, he said: "Fear thou not: (well) hast thou escaped from unjust people."
Said one of the (damsels) "O my (dear) father ! engage him on wages: truly the best of men for thee to emply is the (man) who is strong and trusty."..
He said: "I intended to wed one of these my daughters to thee, on condition that thou serve me for eight years; but if thou complete ten years, it will be (grace) from thee. But I intend not to place thee under a difficulty : thou wilt find me, indeed, if Allah wills, one of the righteous."
He said: "Be that (the agreement) between me and thee: whichever of the two terms I fulfill, let there be no ill-will to me: Be Allah a witness to what we say."