In the 24th century, the European Alliance was a political entity on the European continent on Earth, incorporating the city of Brussels, among others. (TNG: "The Price")
Background information
The European Alliance was most likely a successor to the 22nd century European Hegemony regime, thought to have been the first stirrings of world government on Earth during this time frame. (TNG: "Up The Long Ladder")
Most probably, the European Alliance incorporated the whole European continent, but the possibility also exists that it was only part of the continent. It then would not be clear if this power ever coexisted with the European Hegemony, or possibly shared some other relation to it.
It remains unknown which competences this alliance carried, since Earth was governed by the United Earth Government since the 22nd century. However, the European Alliance might have begun as a joint effort of the European people to create a unified administration and representation at the global government and, thus, might have been very similar to the European Union.
The European Alliance is presumably bordered by the "Mediterranean Alliance", which was referenced in the novelization of Star Trek: The Motion Picture.