In plants where no root grows larger than another, they are said to have a fibrous root system.
 

•  a fibrous root system can have up to 14 million roots, such as in the rye plant
• fibrous roots prevent erosion
• grasses are an example of plants with a fibrous root system

Some plants have primary roots that grow longer and thicker than the secondary roots.  This large primary root is called a taproot.

• taproots make a plant hard to pull from the ground
• taproots can also reach far underground to reach water
• some examples of taproots are dandelions, radishes, beets and carrots