The first and most obvious problem is minority rule. If there are five different candidates and the results turn out like this: (A - 20%; B - 17%; C - 23%, D - 21%, E - 19%), then C would turn out to be the winner. This seems fair at first, until you notice that 77% of the voters didn't vote for candidate C, but wanted somebody else to represent them.
The second problem is the eventual devolving of a winner-takes-all system into two-party rule. Over time, voters lose faith in smaller candidates, not wanting their vote to be "wasted" on a party unlikely to win the majority. This ultimately ends up with two parties controlling politics.
The third problem stems from an attempt to reverse the effects of the second: vote splitting. If a third candidate joins the race, thinking that they may be able to offer a new alternative, they may draw votes from a major candidate with a similar ideology. This is the effects of vote splitting, and it is usually in the opposite candidate's favor. For example, if the first candidate A gets 40% in the final vote, but both second candidate B and our new third candidate C, similar in ideology, each get 30% of the vote, candidate A will win under winner-takes-all, even though 60% of the voters did not want A.
These problems are the main reasons why many consider the winner-takes-all voting system to be flawed. If you are looking for alternatives, perhaps consider instant-runoff voting, or the single transferable vote.