An open cluster is a group of stars that were born at the same time from a molecular cloud, and are still near to each other. They are also called galactic clusters since they exist within the galaxy's disk.
NGC 3603 is home to an open cluster (centre of this image) containing about 2000 bright stars, each of which is much brighter and more massive than our Sun.
Open clusters are usually young (in astronomical terms), and so contains many hot and luminous stars. This makes open clusters visibile from large distances and one of the preferred objects by amateur astronomers. The "parent" molecular cloud is sometimes still associated with the cluster, which illuminates parts of the cloud that are then visibile as one or more nebulae. .
All the stars in an open cluster have more or less the same age and the same chemical composition, so any difference between them is solely due to their mass. Most open clusters are dominated by their O-type and B-type giant blue stars, which are very luminous but short-lived. Analysing the light from an open cluster one can estimate its age, looking at ratio between blue, yellow and red stars. The more blue stars there are, the younger the cluster is. The uniformity between the cluster's stars makes them a perfect test for stellar evolution models, because when comparing one star to the other, most of the variable parameters are now fixed. Testing the model is therefore easier.
The closest open cluster is in Ursa Major, or to be more correct, it is Ursa Major. Most of the stars in the famous asterism are members of an old and mostly dispersed open cluster. Sirius is a former member of this cluster and our sun is in the outskirts of what is called The Ursa Major Stream, a group of stars that are all ex-members of the Ursa Major Cluster spanning over a thousand light years in space. Our Sun is not a member, however, just passing through. Considering its location in the galaxy our Sun has a very strange velocity, we probably had a close encounter with another star a few billion years ago which gravitationally accelerated the Sun and the solar system.
Stars inside an open cluster are at first tighly packed, moving at the same speed around the center of the Galaxy. After half a billion years or so, a classic open cluster such as the Pleiades or the Hyades (both in Taurus) tends to be disturbed by external factors (such as molecular clouds passing by), setting its stars moving at slightly different speeds and so causing them to drift apart exactly like the one in Ursa Major has done. When this happens, the cluster becomes a stream of stars, not close enough to be a cluster but all related and moving in very similar directions at similar speeds.
After a billion or so years, the cluster is totally lost. Some stars will be on the far side of the galaxy, some on the near. The sun's original cluster is like this, there is no way to tell which are former members and which just happened to have formed at the same time but somewhere else.
The exact timeline of this evolution may vary according to the cluster start density: more tighly packed clusters will survive for longer times, but no known cluster has survived more than a few billion years.