Hi
Great question.
In digital imaging each pixel is described by some number values - so, in RGB mode, a pixel might have the following values Red 130, Green 25 and Blue 86. The problem is that the colour described by those values could vary. So they are made to describe a specific colour by putting them in the context of a colour space. So those same values will describe a different colour in sRGB, Adobe RGB, ProPhoto etc. So step one is to ensure the document has an ICC colour profile embedded, in order that the values can describe a specific colour..
Job done? Well not yet. If those values, in the context of a colour space are sent direct to a screen then each screen will display them differently, depending on the type of screen and how the screen controls are set. Just walk into your local TV store to see examples of that. So we also need a description of how colours are displayed by that screen. This is the monitor ICC profile and is usually made by a hardware device along with specialist software.
Job done now? Again , not quite. Our software needs to convert the colours values in the document to the colour values sent to the screen. This is the job of the colour management system. It uses the document ICC profile and the screen ICC profile and converts the values without you doing a thing. Applications that use colour management in this way are described as colour managed and include Photoshop and some web browsers.
As long as you embed the colour profile in your document, anyone using a colour managed system, with a correctly profiled monitor, will view the colours correctly.
So what happens when you send an image to the big wide world? Well for those who care about colour and are using colour managed software they will view your image correctly, provided you embed the profile in the document. For those who do not care, or use colour managed software, they will see incorrect colours - however the usual advice when sending images to the web is to convert the document to sRGB and embed the profile. sRGB will look close to correct on a standard monitor.
What about printing? In the same way as the colour management system converts the colours to the screen, if given an accurate ICC profile describing your printer/paper and ink combination it will adjust the values so that they will print as accurately as possible on your printer (given the limitations of inks).
I hope that helps
Dave