Elections are crucial element in maintain popular sovereignty. They are the mechanism of popular sovereignty. The electorate is mostly ideologically moderate. Thus, according to Democratic Theory, elected officials should be ideologically moderate as well. There should be a connection between preferences of population and who are our representatives. However, in Democratic Reality, elected officials are ideologically extreme. Most of our leaders are ideologically extreme.
The concept of the median voter assumes that voters and candidates on one dimension (Lib/Con) and that liberals are generally Democrats while Republicans are generally conservative. Also, it assumes that voters always prefer candidate closest to their position. Consider a sliding scale with one being the most liberal, ten being the most conservative and five being the median voter. Five is the median voter because there are equal numbers of liberal and conservative voters on either side of the median voter. If you are a four (more liberal) on the scale of liberal to conservative, you will vote for a more conservative six instead of an extremely liberal one.
The theoretical significance of the median voter is that when majority rules apply, the median voter determines the winner. Whomever the median voter votes for, that candidate always wins. If you have a seven or a six running, the six will win. The practical signicance of the median voter is that candidates running for political office usually attempt to attract the median voter.