A colour space is based on a colour model, and is its concrete application to a colouring method (printer, monitor, etc). It therefore contains all colours that can actually be represented through this method and the associated materials.
For example, sRGB and AdobeRGB are both colour spaces based on the RGB colour model. They can only represent a portion of the colours theoretically defined by the RGB model.
The colour mode is a setting that determines the depth to which colours can be displayed in an image file. This also affects the size of the file.
The most common colour modes are: