Taproots are large and fleshy in appearance, grow deep into the ground to search for food and are able to store food, which is particularly necessary for perennial plants during periods of dormancy. They are mostly found in plants with two seed-leaves (dicots).
Fibrous roots lie close to the surface of the soil to collect rain water before it sinks deep into the ground and grows side side with taproots. The two types of root combining efforts to maximize efficiency. They are mostly found in plants with one seed-leaf (monocots).
Adventitious roots develop to help the plant climb: for example, ivies help develop modified underground stems, such as bulbs are common in both dicots and monocots.
Tubers and rhizomes: These include things like potatoes - they form eyes and it is from these eyes that new plants emerge and grow. You can make more of any plant growing from a tuber or rhizome by cutting it into pieces at transplanting time, making sure each piece had a bit of the eye.
Bulbs and corms: Some plants have something more substantial going on beneath the ground - a solid mass of material resembling an onion with roots emerging from the underside. These propagate by splitting off and making more little bulbs and corms that eventually grow to flowering size.