Both the Epic of Gilgamesh and the book of Genesis in the Old Testament contain a story of a major flood and both stories have a male hero that survives the flood, Utnapisthtim and Noah, respectively. Utnapishtim, however, entered "the company of the gods" and came to have everlasting life after he survived the flood. Noah lived to an old age, but his religious tradition allows only one everlasting divinity.
In Utnapishtim's flood story, the gods decide to wipe out the human race because they are too noisy; in Genesis, God decides to destroy humans because of their wickedness.
Both Utnapishtim and Noah are advised by their respective gods to build a boat before the flood comes and they also take their families and have animals aboard the boat with them.
Utnapishtim's rains only last for six days and six nights, whereas Noah's last for 40 days and 40 nights.
Both men release birds three times from the ship to determine whether any dry land has appeared. Utnapishtim releases three different birds (dove, swallow, raven), while Noah releases a raven, a dove, and then a dove again.
The above answer is fairly comprehensive but contains some minor inaccuracies and omissions. Firstly, Utnapishtim was rewarded with immortality, but there is no dramatic contrast here with traditions in the Hebrew Bible. According to oral tradition based on a cryptic comment in Genesis, Enoch was also granted immortality. In Elijah's case this seems even more explicit in the text.
Secondly, we are not told why the gods decide to send the flood in Gilgamesh. Although the Gilgamesh flood story is based on the Epic of Atrahasis - where human overpopulation and noise disturb the rest of the gods - the relevant information is omitted in the eleventh tablet of the Gilgamesh epic. Ultimately, however, the Gilgamesh version contains a similar message to Genesis where the emphasis is particularly focused on "violence" (Gen 6:11-13). The difference is that it is the unwarranted destruction of human life by Enlil, the god primarily responsible for sending the flood, that is condemned in the older, Mesopotamian version of the story.
Other significant parallels include the fact that both ships settle on a mountain range in northern Mesopotamia, both flood heroes offer sacrifice after alighting from their vessels, the deities are attracted by the sweet savour of the sacrifice, a memorial is established (a rainbow in Genesis and an astral necklace that some feel may also refer to a rainbow in Gilgamesh), and the clear implication in Gilgamesh, expressly stated in Genesis, is that there will never be another universal flood.