A taproot is a long, tapering root that goes straight into the soil, which is possessed by many plants, most especially by dicotyledons and gymnosperms. Although it has a long main root, there are other lateral roots that are attached to it, but they are not of the same length as the main root.
Aside from the fact that a taproot gives support to the plant, it can also absorb nutrients, water, and store food. A perfect example of this is what you see in carrot, turnips, beetroot, etc. A fibrous root, on the other hand, has so many roots that are of similar length, commonly possessed by the monocotyledons.
Due to the nature of the fibrous root, it cannot penetrate deep down like in the case of a taproot. In the case of a fibrous root, the primary root that is produced during the early stage of the plant will be replaced by adventitious roots, whereas this is not the case with the taproot.