Lining is used to help hide the inner construction details of a garment, and also to help it slide off and on over other clothing with ease. Lining fabrics are usually slippery and silky, though other types may be used for effect. Lining is constructed separately from the garment and attached at facing or hem areas by hand or machine.
Interlining is a fabric added to a garment when more warmth is needed, like in a winter coat. It may be a heavy fabric with batting added, or a lighter weight one like flannel or fleece. Interlining can be constructed separately from the actual garment (it's sometimes removable), or used as an underlining.
Underlining is a fabric added to fashion fabric for more body and/or opacity. It's a separate layer attached to the corresponding garment fabric section wrong side, and then treated as one during construction. Pattern markings are often transferred to the underlining to avoid show-through on the garment fabric.