"Color management" is a process where the color characteristics for every device in the imaging chain is known precisely and utilized to better predict and control color reproduction. For digital photography, this imaging chain usually starts with the camera and concludes with the final print, and may include a display device in between.
A color space relates numbers to actual colors, and is a three-dimensional object which contains all realizable color combinations. When trying to reproduce color on another device, color spaces can show whether you will be able to retain shadow/highlight detail, color saturation, and by how much either will be compromised.