What is chemistry?

Chemistry is the study of atoms and what they do with their electrons.
Chemistry courses invariably begin with the study of atoms. They do not, however, delve into the intricate workings of the nucleus: that’s the realm of Physics. They begin with atoms as “protons, neutrons and electrons” and you start to learn about electron configurations in shells. You’ll then learn about how electrons bond when they meet other electrons from other atoms, and the properties that these different bonds can give to a material.
In the middle, you’ll learn about , solubility and redox. Electrons are extremely important in all these roles.
Chemistry really “ends” with organic chemistry, which is the study of large, intricate, carbon-based molecules found in living creatures. The study of anything larger than organic molecules (such as membranes or DNA) is usually considered to be the realm of Biology.
This video sums up Chemistry nicely:

And this classic cartoon says it best: http://xkcd.com/435/