Rhizomes are thickened stems filled with nutrients. They feed the plant that will emerge from the rhizome. A rhizome creeps along the ground, growing longer and sending up more growing tips as it grows. This makes rhizomes easy to propagate. You can cut the rhizome into sections as long as each section has roots and at least one growing tip or bud. Bamboo, Cannas, Gingers, Heliconia, and Calla lilies are all grown from rhizomes.
The Ginger you see in the grocery store, "Zingiber officinale," is perfect example of a rhizome we see every day. This is the common "edible ginger" grown for its spicy rhizomes used in cooking and medicines. People will try to buy these at the grocery store and grow them, but most of the time they have been cooled to the point that they will no longer grow.