Star clusters tend to be classified in two ways: by number of stars and by distance from Earth. Open Clusters tend to be relatively close (scores to hundreds of lightyears away) and have relatively few stars (a dozen to hundreds). Globular Clusters tend to be farther away (generally thousands to millions of lightyears away) and have a large number of stars (millions).
An "Open Cluster" is close enough for us to be able to distinguish individual stars. Thus, all "Open Clusters" that have been identified are within our Milky Way galaxy, and many are even part of the same spiral arm as our Solar System. The Pleiades is perhaps the best known such cluster. The Hyades and Trapezium are other examples. These groups of stars are generally young and held together by gravity. They are typically still near or within the gas cloud that spawned them and are therefore relatively close to the same age. Their existence as a cluster does not mean that they will maintain that connection.
A "Globular Cluster" is a much larger and more densely packed grouping of stars. One could think of this as a miniature galaxy. There are two of these near the Milky Way - the Large and Small Magellanic Clouds - neither of which are visible from the Northern Hemisphere. These will probably be absorbed into the main body of the Milky Way Galaxy in the future. Globular Clusters are held together by gravity and stay as such for long periods of time. Many spiral arms surrounding galaxies are Globular Clusters that are in the process of being taken into the larger galaxy.