Star clusters are physically bound systems of stars. In order of low compactness to high compactness (and in some sense also age) they range from stellar associations to open clusters to globular clusters.
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. The most prominent open clusters are the Pleiades and Hyades in Taurus, but h+Chi Persei can also be a spectacular sight outside the glare of human settlements.
The brightest globular cluster is Omega Centauri, visible with the naked eye from the southern hemisphere. The best known globular cluster in the northern hemisphere is M13 (modestly called The Great Globular Cluster) in Hercules.
An open cluster is exactly that: A relatively loose collection of stars. Stars are normally separated 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 separated 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. Mirfak (&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 separated. In a few million years, they will be a moving cluster. They were both formed at around the same time (give or take a hundred million years) and in the same manner. A cloud of gas, like M42 today, begins forming stars. Since the cloud is of a finite size then the stars will form close by each other. Eventually they drive away the remaining nebula and appear as another open 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. Others were formed in much the same way as open clusters, but on a much, much grander scale. There is a low mass globular currently being formed in the SMC, astronomers use this as a unique insight into the formation of these objects since no other nearby sites like this are known. When a globular is forming, it has to form from a molecular cloud and a relatively dense one. These are rare, and explain why a galaxy has far, far less globulars than it does galactic clusters. Also, the rate of condensation of the molecular cloud must be low. It's no good forming massive O or B stars because they'll just explode after a few million years. The majority of stars that have to be formed are those of the classes similar to our sun, F, G and K. These will burn steadily for billions of years and not be so violent as to evaporate the nebula before the cluster has grown to the size needed to be a globular. Lastly, the globular, being very dense compared to other regions of space, will be ejected from the plane of the galaxy and enter a stable halo orbit.