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Mythology is the study of the stories and legends that inform a religion or culture about its laws, its history, its morals, its social constructions, and really everything else that define the culture or religion's world view. The term "mythology" is also used to define the set of myths that inform any particular religion. Basically what we believe is religion or theology while what those others believe is legend or mythology.
All religions have myths and associated mythologies; however, some members of contemporary religions (in particular, Christianity and Islam) expect their religions not to be held to the same standards as other religions when studied in a non-theological setting, expressing the notion that their holy stories should not be termed "myths".
Despite the popular idea of myths as being something "false", neither truth nor lack of truth defines a myth. What makes a myth is how it impacts upon the culture and the culture's passing on of its standards.
 Study of mythology
The study of a culture's mythology provides historians, anthropologists, sociologists and other scholars insight into world view of that culture. For example, Coyote's sexual exploits in Lakota mythology would suggest to scholars that the Lakota were open about sexual encounters and accepted the humor inherent in sexual situations. On the other hand, Eve's shame at her naked body would indicate that the ancient Jewish communities were more reserved about exposure of the body and about expressing sexual interest openly.
Parallel mythologies can be used to demonstrate who has influenced whom in the development of religion and culture. They can also demonstrate the distinctions in an individual religion's understanding of similar concepts shared by two or more religions. Though they tell the same story of the Garden of Eden, the two formative scrolls of Bereshit have fairly different images of God. As seen in the Adam and Eve story, one scroll (YHWH-Elohim) defines a highly personified god who is "heard" walking through the gardens, who "calls out" to the couple he has "lost" and cannot find. He "guesses" by their modesty, that they have eaten of the fruit. This god is not perfect nor all knowing. In the second scroll, however, YHWH is far less personified, acts from "on high", more indirectly and does not demonstrate the more "human" emotions of surprise and disappointment.
 National mythologies
Not all mythology is associated with a religion. The United States, for example, has a very particular mythology about its place in the world that is seen through myths of the American Dream and of America as a supporter of human rights, freedoms and democracy. This is part of a system of beliefs that scholars of religion have termed American Civil Religion.
National mythologies are an excellent source of pseudohistory. Some famous national myths include: George Washington and the cherry tree; that the Revolution was won, simply cause we are "better" (largely overlooking the importance of other world conflicts of the day).
 Mythology of Science
Mythology of Science is also unrelated to religion as it involves scientific discoveries and/or inventions and the people behind them, such as the story of Isaac Newton and the apple. Notably, the one about Christopher Columbus on RationalWiki's scientific mythology page would also fit under the previously mentioned American mythology.
Scientific mythology is also prone to contain pseudohistory.