A taproot is a large main root that comes off of the stem and has many smaller lateral roots; a fibrous root system has many roots of the same size that break off into small lateral roots. Dandelions, poison ivy and carrots are all examples of taproots; sweet potatoes are the fleshy part of a fibrous root.
The taproot system allows plants to anchor better, get sources of minerals and water from further into the ground than fibrous rooted plants. The fibrous system is more susceptible to drought; however, it also allows the plant to respond faster to an application of fertilizer.
Dicots and monocots are the two classes of flowering plants. The majority of taproot systems are composed of dicots and conifers. The taproot is the main root in the plant, but there are smaller roots that grow out from the root. A tap root grows downward into the ground. For this reason, carrots are only picked out of the ground. Taproots are able to dig deep into the ground to reach reserves of water to keep themselves sustained.
Monocots, including onions and grasses, are fibrous root systems. The roots of these plants are somewhat small but are generally around the same size and grow out of the stem. From the main roots, smaller lateral pieces of roots grow outward and begin to plant themselves in to the ground. The smaller lateral roots are able to attach and cling to the soil, which makes them ideal for the prevention of erosion.