In the culinary arts, anise (pronounced "AH-niss") refers usually to the seeds of a plant with aromatic leaves and stems that taste like liquorice, fennel or tarragon.
While the leaves of the anise plant can be used as an herb, it's mainly the seeds that are used in cooking. Sometimes called aniseed, the seeds are used as a spice, either ground or whole. Aniseed is used in a various baked goods and desserts, for example Italian biscotti. Anise seeds are also frequently used in making sausage.
Anise seeds are also the basis for a number of alcoholic beverages, including absinthe, anisette, ouzo and sambuca.
Despite its similar name, anise is not related to star anise, which is another spice from a different family of plants.
Nor is it the same thing as fennel, although the two do have a similar flavor, and the plants are somewhat similar looking. They're from the same family of plants (along with caraway, parsley, cilantro and others), but they're not the same species.
In general, fennel is served as a vegetable, while anise is used as a spice (i.e. in seed form, either whole or ground).
Also see: Quiz: Is it an Herb or a Spice?