These lines at the beginning of the poem Paradise Lost by John Milton comprise a summary of the Biblical story that the poet is telling: an account of Adam and Eve in the Garden of Eden, Satan's temptation, and the fall of man. It serves as an introduction to the poem that is to follow. You can find this account in chapters 2 and 3 of the book of Genesis in the Bible.
"Man's first disobedience" refers to the first time that Adam and Eve deliberately go against God's commandment. This has to do with "the fruit of that forbidden tree." When God creates Adam in the garden, he tells him that he can eat anything he finds there except the fruit from the tree of knowledge of good and evil. God warns Adam that if he eats from that tree, he will die.
God then creates Eve, and evidently Adam has told Eve what God said, because when Satan talks to her, she is aware of the restriction against eating from that one tree. However, Satan tempts her to eat it, and she does, and then she coaxes Adam to eat it as well. This is the fruit "whose mortal taste brought death into the world, and all our woe." In other words, because Adam and Eve taste the fruit from the tree of the knowledge of good and evil, they become disobedient, something that has never happened before. As a result, God keeps his word and drives them out of the Garden of Eden, which is like paradise, and into the world outside, where they have to work hard, suffer, and eventually die.