What is Abraham's relationship to God?

Answer 1

Abraham was a prophet of his God, which is kind of a fancy way of saying Priest/rabbi/whathaveyou.

Abraham was a deep believer in his God, so he went around preaching the word of Him. He's the Jesus of Judaism, but he never claimed to be the son of God (in the Christian sense, anyway).

Answer 2

God had a close relationship with Abraham and promised him his descendants would ultimately comprise many nations. God even changed his name from Abram to Abraham, meaning "father of a multitude," to reflect the importance of this promise.

Answer:

As stated in the above answer, God expressed His esteem for Abraham (Isaiah 41:8). Here is some more background:

According to tradition, Abraham founded Judaism, and Moses later received the Torah from God.
Abraham, tenth generation descendant of Noah, of Hebrew lineage, was the son of Terah, uncle of Lot, father of Isaac, grandfather of Jacob, and ancestor of the Israelites. His story is in Genesis ch.11 (end), through ch.25. Jewish tradition states that he was the first to teach belief in One God; and it is in his merit that Jews continue to exist (Genesis 18:19, and ch.17).
Abraham (18th century BCE) came from ancestry that had been God-fearing a couple of centuries earlier but had afterwards slipped into idolatry (Joshua 24:2). Nimrod, the idolatrous tyrant, had brought Abraham's father (Terah) from the Semitic ancestral seat near the conjunction of the Balikh and the Euphrates, and instated him in a position of power in his army in the royal Babylonian city of Ur, where Abraham was born. Nimrod persecuted any who would question his idolatrous cult.
The Kuzari (Rabbi Judah HaLevi, 1075-1141) states that Abraham was gifted with high intelligence; and, as Maimonides (1135-1204) describes, Abraham didn't blindly accept the ubiquitous idolatry. The whole populace had been duped, but the young Abraham contemplated the matter relentlessly, finally arriving at the conclusion that there is One God and that this should be taught to others as well. This is what is meant by his "calling out in the name of the Lord" (Genesis ch.12). As a young man, he remonstrated with passersby in public, demonstrating to them the falsehood of their idols; and our tradition tells how he was threatened and endangered by Nimrod.
Subsequently, Terah relocated to Harran; and it is here that Abraham began to develop a circle of disciples (Rashi commentary, on Genesis 12:5).
Later, God told Abraham in prophecy to move to the Holy Land, which is where Abraham raised his family.
He continued his contemplations, eventually arriving at the attitudes and forms of behavior which God later incorporated into the Torah given to Moses. Abraham taught disciples (Talmud, Yoma 28b), gave tithes (Genesis ch.14), strove to raise a family (Genesis ch.15, 17, and 24) which would serve God (Genesis 18:19), made a covenant with God (Genesis ch.15 and 17), welcomed guests into his home (Genesis ch.18) unlike the inhospitable Sodomites (Genesis ch.19), prayed for people (Genesis ch.18), rebuked others when necessary (Genesis ch.20), eulogized and buried the deceased (Genesis ch.23), and fulfilled God's will unquestioningly (Genesis ch.22). He became renowned as a prince of God (Genesis 23:6).
All of these forms of behavior were based upon the ways of God, which Abraham comprehended through his contemplations. These, and similar personality traits, were the teachings of Abraham and his descendants.
It is therefore clear why God expresses His love for Abraham (Isaiah 41:8) and calls Himself the God of Abraham (Genesis 26:24), and says that Abraham obeyed Him fully (Genesis 26:5). And this is why, according to our tradition, Abraham is credited with having begun the religion which became known as Judaism. However, Abraham and his descendants observed their traditions voluntarily, until the Giving of the Torah to Moses 3325 years ago, when God made it obligatory.