On the simplest level, the roots are anything below the ground and the shoots are anything above the ground.
On a more technical level, the root system includes the roots (of course), tubers, and rhizomes. The shoot system includes leaves, buds, flowers, stems and fruits, if applicable.
The root system is very important, because through the root system a plant or tree gets water and necessary nutrients for growth to take place. Without a good root system a plant or tree will not grow to its potential or even die. Moreover, the roots of a plant or tree provide the necessary anchorage.
The shoot system is also essential for the proper growth of any plant, because it is through this system that photosynthesis takes place. Through photosynthesis, sunlight, water, and carbon dioxide is converted into oxygen and sugars, which are needed for growth.
to answer that in a more technical way, the shoot has 'repeating modules' or something like a pattern of nodes, leaves, buds on it, as well as things like axillary buds, that goes something like "node + internode + axillary bud + leaf"
axillary buds are buds that grow on the node, which my teacher calls 'that part that looks like an armpit'