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.
See this thread for comments on astronomy vs. astrophysics.
Cosmology is a subtype of astronomy and astrophysics which pertains to the universe as a whole---as cephron's wiki snippet suggests. I don't think cephron's comment about phenomenology is correct, however. Cosmology is a huge aspect of observational astronomy---particularly the study of galaxies, and radio/microwave observations of gas in the distant universe.
Common topics in cosmology include: the big-bang (and inflation), nucleosynthesis in the early universe, galaxy formation and evolution, dark matter and dark energy, etc.
In general, I would say that Astronomy is concerned with everything outside the Earths atmosphere, from the Moon, to the Sun, to other galaxies. Astrophysics is a sub field of Astronomy, as is Cosmology. Astrophysics doesn't care about the time that Virgo rises unless you are asking about the physics of the rotation of the Earth and it's orbit around the sun. Similarly cosmology doesn't care about the magnetic field around Jupiter influencing it's moons, which would fall under Astrophysics. However it would care about the impact of supermassive black holes on galaxy formation, as would Astrophysics and Astronomy in general.
I would say you can't be a competent cosmologist without deep knowledge of astrophysics, but, you can be an outstanding astrophysicist without knowing squat about cosmology. I also agree the term 'astronomy' subsumes both astrophysics and cosmology. Astrophysicists tend to be the most specialized.