Each output device, for example, your display or printer, is only capable of producing a certain range of colors. A color space is a specific implementation of the color model used to define the color gamut (i.e., the range of available color). For example, Adobe RGB, sRGB, etc, are all unique color spaces for the RGB color model. Different color spaces are also available for CMYK and Lab color models.
In order for a device to know which color space to use, it looks at the assigned color profile. You can choose your color space by assigning a color profile to your document.
Which color space you choose depends on what you're doing and the color model you're choosing to operate in.
If you're unsure of what color space to operate, it's advisable to stick with the default sRGB IEC61966-2.1 profile if using the RGB color model.