We define certain special elements of bisemilattices, such as distributive, neutral, absorptive, and so on. After giving some properties of these elements we consider two kinds of ideals and filters of bisemilattices and their connection with special elements. We prove structure theorems for bisemilattices as generalizations of the well known structure theorems for lattices. Finally, we give an application related to bisemilattice identities satisfied on a bisemilattice and a consequence of this result for lattices are given.