Midway Island is a small atoll that is located in Earth's Pacific Ocean, specifically that region known as the Tropic of Cancer. It is near to the American state of Hawaii, the atoll of Wake Island, and is located "midway" between the east coast of North America and the islands of Japan, in Asia.
Midway Island was not mentioned by name but can be seen fairly clearly on the map downloaded from the Enterprise computer in "The Cage". On the map, Midway Island appears as "Midway Is. U.S."
The Battle of Midway took place near to Midway Island, and was won by the American Navy. It was considered a major turning point in World War II which ultimately led to the defeat of Japan in 1945. The USS Enterprise was among the aircraft carriers present at the battle, along with the USS Hornet and the Japanese carrier Akagi (which was actually sunk by aircraft from the Enterprise). In TNG: "Redemption II", the USS Hornet and USS Akagi were both part of the fleet which formed the tachyon detection grid; Ronald D. Moore intended this as a symbolic gesture of Humanity's willing unification that two starships whose namesakes had fought one another would now serve side-by-side.