Reproduction is the creation of a new organism from an existing organism or organisms. Two forms of reproduction exist: sexual and asexual. Sexual reproduction involves the combination of genetic material from two parent organisms to create a new one. Asexual reproduction is the duplication of the genetic material from a parent organism to create a new organism. Mitosis is the way in which asexual reproduction takes place, but it takes place in many other contexts as well.
Living cells often get to a stage at which they need to divide. This happens with stem cells in your bone marrow, muscle cells in a frog's leg and yeast in your loaf of bread. A cell cannot survive without DNA to direct the synthesis of proteins, so when a cell divides, both parts need to get DNA. The first step in cell division, then, is creating an extra copy of the cell's DNA and splitting it up. The duplication and division of DNA is called mitosis, and it's quickly followed by a step called cytokinesis, whereby the cell tightens a belt around its middle and splits into two daughter cells.
Reproduction is the synthesis of a new organism from an old. The trillions of cells in your body are all the result of many cycles of mitosis, starting from when you were a single-celled zygote. All those cell divisions were not reproduction, though, because you're still a single organism. So mitosis is not necessarily reproduction -- but it can be.
Eukaryotes are a type of cell that keep their DNA enclosed within a nucleus. The DNA is also organized as a set of long linear molecules called chromosomes. When a eukaryote is ready to divide, the chromosomes duplicate, the nuclear membrane dissolves and the separate copies of the chromosome get dragged to opposite sides of the cell before it divides. The cell duplicates only its own DNA -- it doesn't get DNA from any external source, so its reproduction is asexual. But mitosis isn't the only form of asexual reproduction.
Other Asexual Reproduction
Prokaryotes are single-celled organisms with no nucleus. Their essential DNA is in the form of one main loop of DNA that forms its single chromosome. Bacteria, for example, are prokaryotic organisms. When a bacterium is ready to divide it will duplicate its chromosome, but there's no nuclear membrane to dissolve and no active movement of the chromosomes from one side to another. Even so, after the chromosome duplicates, the cell splits into two daughter cells. No exchange of genetic material has taken place, so this is a form of asexual reproduction -- without mitosis.