Star clusters are held together by the gravitation of their members. Due to both external (encounters with massive objects, influence of the host galaxy) and internal (encounters with other cluster members, stellar evolution) influences, clusters slowly evaporate. Their lifetime varies from a few million years for loose associations to many billions (milliards) of years for massive globulars.
The brightest and nearest stellar clusters are visible with the naked eye. Quite prominent are the Pleiades and Hyades in Taurus, but h+Chi Persei can also be a spectacular sight outside the glare of human settlements. All of these are open clusters. The brightest globular cluster is Omega Centauri, visible with the naked eye from near the south celestial pole (i.e. from the southern hemisphere).
The Hyades cluster still serves as an important stepping stone on the cosmic distance ladder. In general clusters, due to their rather homogeneous stellar population and relatively well-known distances, play an important role in astrophysics and astrometry.
An open cluster is exactly that: A relatively loose collection of stars. Stars are normally seperated by a few light years although they can be closer. Eventually, interactions and the slightly different orbits of the stars around the galactic centre will force the open cluster to disperse. Large open clusters will form streams, vast groups of stars sharing a common motion through space but seperated by tens of light years. Our Sun is in the Ursa Major Stream at the moment, but isn't a true member, just passing through. Also at this stage, the most tightly bound members of the former open cluster will still be close together but not close enough to be called an open cluster. This is a moving cluster, most of the stars in Ursa Major are members of the Ursa Major Moving Cluster. This was once the core of the Ursa Major Stream.
Alpha Persei is the lead star in the Alpha Persei Moving Cluster, another easy moving cluster. Distant moving clusters can't readily be detected since the proper motions of the stars need to be known.
Of the well known clusters, the Hyades and the Pleiades, the Hyades are very nearly a moving cluster, being relatively sparse and well seperated. In a few million years, they will be a moving cluster.
Globular clusters aren't found in the plane of the galaxy, but all around it. Many of them are probably the cores of galaxies long since merged into ours, having had all their gas, dust and dark matter stripped from them until only the denser stars remain.