A core world is one that you assign a governor. This allows you to open the planetary view screen and build things on the planet tiles, as well as grow population.
A colony is a world that you do not assign a governor, you don't get a planetary view screen to build things on the world, it's population does not grow beyond the one used to colonize it. It sends all of it's resources (the numbers for minerals, technology, wealth, food, and influence) the the nearest core world.
Basically a colony adds is base values to the core world it is sending resources, minus whatever decay percentage it has based on distance to said core world.
That is the best way I can describe it in a short amount of time.
Happy Gaming !