I'm sure a more complete treatment of the subject could be given, but I'll start the bidding with Wikipedia's first paragraph on each:
Astronomy is a natural science that deals with the study of celestial objects (such as stars, planets, comets, nebulae, star clusters and galaxies) and phenomena that originate outside the atmosphere of Earth (such as cosmic background radiation). It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the formation and development of the universe.
Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior. Among the objects studied are galaxies, stars, planets, exoplanets, the interstellar medium and the cosmic microwave background.
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order.
My attempt at an analysis:
-Astronomy's focus is objects and phenomena; it is more concerned with particulars.
-Astrophysics is a subdomain of Astronomy which ties in more directly with physics. It is more concerned with universals than particulars, but uses particulars (eg. Black holes) as case studies.
-Cosmology is primarily concerned with universals like space, time, gravity, and fundamental constants, and the study of phenomena is only interesting inasmuch as it informs the understanding of these.