Bean sprouts are a common ingredient, especially in Eastern Asian cuisine, made from sprouting beans. The typical bean sprout is made from the greenish-capped mung beans. Other common bean sprouts are the usually yellow, larger-grained soy sprouts.
If you are allergic to beans, you might be allergic to bean sprouts. In the event of an allergic reaction after consuming bean sprouts, contact your allergist.
Bean sprouts are typically found in soups or mixed into stir fries. They may also be used as an ingredient, such as in spring rolls (Wikipedia, nd).
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Here is an interesting video tutorial describing how to grow your own bean sprouts!