Most universities structure their courses in a gradual concentration format. For example, at the University of Otago a history major has to take two history courses in his first year, then four in his second year and four in his third year. If he chooses to do an honors degree, his fourth year consists of nothing but history classes, in addition to a thesis on a specific topic.
This gives students some flexibility. A student could theoretically declare one major in his first year, but meet the requirements for more than one major at the first-year level. This allows him to easily change in his second year without having to go backwards and study another first-year class.
The minor is similar to the major, except it requires less coursework. Whereas a major may require two classes at the first-year level and four at the second-year level, a minor might only require one of each. This is because your minor is a secondary specialization -- you are not as specialized in the subject as you are in your major subject, but you will still graduate with more experience than someone without a minor in that subject.
Minors depend on majors; you cannot usually graduate from a university with a minor but without a major (although triple minor programs do exist). On the other hand, though, you can often graduate with a major, or even a double major, without a minor; majors can exist without minors, but not vice versa.
A major is usually the first step towards qualifying for further study in a field; a minor, on the other hand, is not. So, if you have an English major with a history minor, you would have an easier time getting into a master's program in English than in history.. This is not to say that getting into a graduate history program would be impossible, but you would have to demonstrate through other means that you have the necessarily high level of base knowledge. If, however, you wanted to study biochemistry at the master's level, you would have a hard time with the undergraduate combination mentioned above.