I don't know in Islam, but in Christianity, they were 2 distinct people as far as I know - both sons of Abraham by 2 different women.
Abraham had a wife named Sara (I believe). She was not able to have children so she allowed him to concieve with a slave girl named Hagar. Hagar's son, Abraham's first, was named Ishmael and was the father of the arabs. Sara later bore a child named Isaac who by tradition was the father of the Jews and later Christians. The Eid of Sacrifice represents Abraham who was asked by God to sacrifice his only son (at the time) Ishmael. He told Ishmael, who agreed, and during the sacrifice a Goat appeared instead (by traditional stories). The sacrifice of Abraham is marked and celebrated during this time by Muslims. They are in turn to sacrifice an animal for God and give the meat to charity. Hope that helps.
Tradition also states that Hagar and Ishmael were cast into the desert by Sara and had no water. They ran around an area several times (which is replicated during the Pilgramage to Mecca) until water sprouted up. This became the spring of Zam Zam which is still in existence today in the city of Mecca in Saudi Arabia.
