Color management works by utilizing profiles of each device involved in the workflow in order to guide the process of translating color from one device to another. Color is not the same in every context, sometimes colors are reproduced with RGB elements and other times they are reproduced with CMYK elements (or even more colors than that, as is often the case with ink jets).

In order to accurately understand how color is produced from a source device (say a camera), and to accurately reproduce color on a printer, one must know the characteristics of those devices. Furthermore, in order to see color accurately while you are processing photos, the color management system must also know the characteristics of your computer screen.

Therefor, every device or component involved in a workflow must be identified by it's own profile. You cannot use a profile intended for use with an image (such as sRGB, AdobeRGB, ProPhotoRGB) to calibrate your screen...the screen has different characteristics. Therefor, you must produce an ICC profile for your screen, that explicitly describes its unique, inherent traits. Now, the color management system can reference say the sRGB profile your image is tagged with, and reference the ICC profile of the screen, and perform the necessary translation when rendering that image to your screen, thereby producing accurate color reproduction.

It's no different with printers. Printers are not computer screens, therefor they cannot be properly calibrated by using an ICC profile for a screen. Neither are they images, therefor they cannot be properly calibrated with the sRGB or AdobeRGB profiles. Printers must have their own profiles. With printers, it is a little more complex, because you have not only the device, but also it's inks and the papers you print on. All three of these aspects combine to require a unique ICC profile for each paper, not just the printer & ink. Hence the reason you have to find and use exactly the correct ICC profile for the type of paper you are printing on.

When it comes to softproofing, a tool like Photoshop will utilize all three profiles. It will use the image profile (i.e. sRGB) to calibrate a translation into "printer space" via the printer & paper's ICC profile. Then it will render that translated result to the screen via the screens ICC profile. This is the only way you can accurately soft proof a photo and determine if it will look as you expect in print on the chosen paper type.

Profiling for print can be even more complex. If you have a specific lighting context within which you wish to view a print, you often need to either recalibrate under that specific light, or apply a mathematical transformation to the ICC profile after it's been generated in order to correct the print color so it renders properly under the chosen illuminant. Modern higher end printer calibration systems usually offer features to support generating or using printer/paper ICC profiles for differing illuminants.