What was relationship between snape and albus dumbledore?

Answer 1

Snape was a student at Hogwarts along with James Potter, Sirius Black, Remus Lupin, Peter Pettigrew and Lily Evans (to name a few significant others in the story). Whether or not Hogwarts was under the ruling of Albus Dumbledore or not I am unsure however if Dumbledore was not headmaster then I can assure you that he was a professor at the school.

Later when he left school, Snape became a Potions master at Hogwarts. He had a deep love for Lily Potter (nee Evans - Harry's Mother) and a deep hatred for James Potter (Harry's Father). When he became potions master I am positive the school was under the ruling of Albus Dumbledore.

The two became closer than some of the other teachers at the school largely due to Snape's learning of Voldemort's plans with the Potter family. Snape went to Dumbledore to seek protection for Lily - reluctantly giving in to protection for the three Potters, one of which he hated and one of which he didn't know. Snape was then angry when the protection provided did not end up being enough and the love of his life died.

Snape also played an important part in the whole plot. He was Dumbledore's assistant, of sorts. He was meant to kill Dumbledore to ensure the safety of the Elder Wand. He was also a part of the Order, an organisation of sorts that Dumbledore set up to plot against Voldemort. He was enlisted to deliver the sword to Harry and to tell Harry the truth when the time came.

In short (though this is a very confusing subject to describe in a nutshell) Snape was a student and then teacher under the ruling of Albus Dumbledore, and the two became rather close.

Answer 2

They both were at Hagwarts together, and Snape was left out of Lupin's group of friends; James Potter, Sirius Black, and Lupin.