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Galaxies have gravitationally bound collections of stars in them. One type of these collections is called a globular cluster. Globular clusters contain hundreds of thousands of tighly gravitationally bound stars. They normally exist near the center of a galaxy and orbit its core.They contain relatively old stars, and their formation is not well understood.
Open or galactic clusters are groups of 10s, 100s or sometimes 100s of stars that were born from the same initial cloud of gas and dust. When they are young (a few million or tens of millions of years old), these clusters contain some very large, bright stars (called O or B-type stars). The very youngest clusters often still contain the remains of the gas cloud from which the stars were born and this is seen as nebulosity.
Cluster stars are very useful to astronomers because they were all formed from the same giant cloud so they have the same chemistry. They are also all at about the same distance from us. When observing a group of stars in a cluster, astronomers can assume they are all made of the same stuff, and they are all the same distance away from us – so any differences between them are really caused by their different mass.
As we look from Earth towards a cluster, it's easy to think that we're looking through empty space. In fact, it's not as empty as all that. There can be clouds of gas and dust along the line of sight, which obscure the light from all the stars behind the cloud somewhat. The clouds tend to absorb bluer colors of light the most, causing the light from the background stars to appear more red than they otherwise would. This extinction effect is therefore called "reddening". This changes the value of the star's color indices, B-V, R-I etc.
So imagine you're looking towards a cluster. There will be other stars along the line of sight - not cluster members - both in front of and behind the cluster. And there may be clouds of dust along the line of sight as well. So the light from the cluster stars will all get the same degree of reddening. The light from any background stars behind the cluster will get reddened by the same degree at least and probably more (being further away, more dust between us and them). But the light from the foreground stars will be reddened less.
Therefore, the colors of the cluster stars will change, but they'll remain as a clump together on the color-color and color-mag diagrams, whereas non-member stars will have their colors shifted to various degrees, and they will fall in different parts of the diagram.
In this way, color-color diagrams can help us to decide which stars in our field of view are members of the cluster. But it's not a certain method because it's hard to quantify exactly how much reddening is going on (the distribution of dust in the galaxy is not uniform). If it's a field of very low reddening (little dust along the light of sight), then this technique will be less effective, and we have to resort to other methods to decide membership. The only way to decide membership for certain is to measure the motion of the stars in space over time, but this is much more demanding observationally. So color-mag and color-color diagrams are used to assign a probability that a given star is a member.