By way of Andrew Sullivan, a succinct explanation of the failings of democracy run as majority rule. My favorite analogy: A group of friends wouldn't force the lone vegetarian to eat pepperoni on their pizza.
Why not majority rule? Three general reasons. Madison's reason was what he called tyranny. To a large extent, Madison's concern is pragmatic. He's aware that up to 1776, republics (and forget what someone told you; we're best off following the political theorist Robert Dahl and treating "democracy" and "republic" as interchangeable, really) were famous for being short-lived and unstable. Why? Because losers in a true majority-rules democracy, faced with a true majority, have no stake in the survival of the system and welcome a change. So one wants instead a system in which losers do not lose everything, and have a reasonable chance of winning in the future. To see the intuitive case for this, consider a situation in which you and a small group of friends need to choose an activity or a pizza topping or music to listen to. Perhaps, of the activities you all like, four of you prefer bowling, while two prefer miniature golf. Bowling, of course, "wins." And it wins the next time. After a while, however, odds are that you're all going to agree to let the miniature golf fans have a turn. That's not a violation of collective decision-making among equals; it's a consequence of it, in a situation in which it's easy to discuss fair outcomes. So even in a democracy, it's possible that the same group shouldn't always win.
We can also describe a situation in which the correct democratic solution should probably be that the minority wins: when an intense minority is opposed by an indifferent majority. I always use pizza toppings for this one. If one of your friends is a vegetarian, you're not going to insist on outvoting him and getting the pepperoni, and you're certainly not going to outvote the friend who is allergic to mushrooms. As her friend, you would realize that's just not right, even in milder versions in which, say, three of you have a very mild preference for anchovies, but the other two hate anchovies (imagine some decision rules that force such decisions; you can't cheat in this hypothetical and order a second pie, or whatever). And again, I'm going to strongly argue that two-beating-three in this situation isn't a violation of democracy, but a real, correct, democratic solution.