Bulbs are the thickened underground storage organ of the group of perennials which includes daffodils and tulips.  True bulbs, if cut in half vertically, reveal the components you would expect to find in a bud, namely, flower and leaves. Bulbs are layered much like an artichoke. Onions, tulips, lilies, and narcissus are true bulbs.

Corms are thick, enlarged underground stems which produce the new roots, leaves and flowers during each growing season, examples are crocus and gladiolus.

A rhizome  is a modified, swollen  plant stem which grows horizontally under the surface of the soil and send up leaves and flowers at intervals.  New growth emerges from different points of the rhizome. Irises are rhizomatous plants.

The term 'tuber' is applied to any plant with underground storage parts that don't fit the above categories. There are different kinds of tubers.  Potatoes are considerd a tuber.   Tubers are flat underground stems which store food and plant energy and from which a plant grows.  Dahlias are considered a tuberous root.