Colour Model vs. Colour Space
A colour model is a general system for assigning numbers to colours. The purpose is to have some kind of standard notation with which to designate colours. An example is the RGB colour model, where each colour is defined in terms of three numbers that represent the amount of each pure colour in a mixture that results in the colour you see. Each number represents the amount of the red primary, green primary and blue primary colour.
A colour space is an instance of a colour model in which every colour is represented by a specific point in space, so that a specific set of three or more numbers can describe it. A well-known colour space is the RGB space of your monitor, where a certain colour has a specific set of three numbers to represent the amounts of red, green and blue elements in the LCD matrix needed to produce that colour.
Device-dependent Colour vs. Device-independent Colour
If you want to represent the purest red, you're probably thinking about red that in Photoshop RGB is notated as R=255, G=0, B=0. This purest red will visually differ depending on whether it is interpreted by a CRT-monitor, a LCD-monitor, a scanner, or a digital camera. For example, the scanner will 'see' a brighter red than your LCD monitor is capable of showing you. The difference between the visual qualities of colours depending on the device used for interpretation, is called device-dependent colour.
Device-dependent colour requires all devices in the workflow to be "calibrated" to obtain consistent colour reproduction. Calibration ensures the device is put into a known state, so that it becomes possible to describe the way it interprets colours. A calibrated device can be characterized. Characterization in turn allows devices to be tuned to each other. In other words, the input devices (for example scanners or digital cameras), monitors and output devices (for example digital colour printers) have to be tuned to each other to make the colours match.
Differences may occur between devices of the same type or model, because they may have different settings, or the quality margins for manufacturing the devices were not tight enough. Some very expensive monitors are manufactured to such tight standards that it is possible to have two of them with only the slightest difference in colour representation. However, most monitors under the 5,000.00 USD mark will differ considerably. Printers of the same brand and model will almost always differ, especially inkjet printers.
(Cheap) Inkjet printers in particular are bad at keeping a consistent colour quality through time. They are also bad at maintaining a consistent quality in the representation of a gray ramp (from white to black). This behavior is called 'drift', and calibrating the printer at regular intervals therefore is a requirement for efficient colour management.
Calibrating is almost always an activity internal to the printer, usually set up by the manufacturer. In the cases of HP and Epson, such calibration is called a 'closed loop calibration' as it involves the printer self-controlling its calibration results. The process involves the device printing a number of sheets with colour patches and lines that are checked and acted upon by a built-in measuring and compensation device.
The characterization of a device is also called 'profiling'. Modern operating systems and image editing applications use the International Color Consortium's (ICC) colour management model that has ICC colour profiles at its core. Profiles that are custom-created for the device at hand tell applications how to compensate for the specific interpretation of each colour by the device. Most manufacturers of monitors, scanners and printers deliver standard profiles with their devices.
To eliminate, or at least minimize, colour differences between devices, the ICC colour management system model uses a standard, device-independent colour model --one that is based on how humans perceive colour-- such as CIELab. An ICC profile will bridge the difference between device-independent colour and device-dependent colour, and it will bridge the difference between how humans interpret colour and how the device does.
To humans, the (visible) spectrum contains millions of colours, but colour devices, such as scanners, monitors and colour printers can only (re)produce a subset of this spectrum. This subset is called a colour gamut. The gamut of a device defines the colour space it can (re)produce. For example, a monitor can display a wider range of colours than an offset press can print using CMYK colours, while some of the CMYK colours cannot be accurately displayed on the monitor. Each device has a different colour gamut. In many cases, this may also imply that your printer is capable of outputting colours you can't see on your monitor, and vice versa.
To accommodate for differences in device gamuts, colour spaces like AdobeRGB and ColorMatch serve as go-between. These are large colour spaces that are not tied to any particular device, but serve as a translation 'medium' for the conversion from one device's colour gamut to another.