The Framers believed their most important action in preventing the rise of tyranny in America was to divide the key powers of government among the three branches. They could have simply stopped there, leaving it up to the officials of the three branches to figure out how to defend their own powers and limit the powers of the others. But that might have led to a chaotic series of bare-knuckle showdowns between the branches. So the Framers carefully constructed a system that provided specific levers of power to allow each of the branches to influence the actions of the others in an orderly and predictable way. Those levers are the system of checks and balances.
The core idea of the system of checks and balances was that no one branch of government should be able to get too far out of control without being put in check by the others. If a president starts trying to act like a despotic king, he can be impeached by Congress. If Congress starts trying to pass a series of laws that are blatantly unconstitutional, those laws can be overturned by the Supreme Court. And so on.
The most important result is that getting anything important done within the American system of government typically requires the cooperation (or at least the acquiescence) of more than one branch of government.
For example, the president is named by the Constitution as commander-in-chief of the armed forces, giving him an incredibly powerful position in times of war. The Framers worried that the president's wartime role was too powerful, in fact, and thus gave Congress a powerful set of checks and balances on the president's war powers. Only Congress, not the president, has the power to declare war. Perhaps even more important, only Congress has the power to pay for wartime expenses. That means that if the president tries to launch an ill-advised military escapade, Congress can effectively pull the plug, forcing the president to bring his troops home by refusing to fund their continued deployment.
Another example: the president can veto acts of Congress. Thus if the executive tries to pass laws that the president believes are unconstitutional or even just bad policy, the prez can whip out his red veto pen and prevent the bill from becoming law. The veto is perhaps the president's most powerful means of checking and balancing Congress… so powerful, in fact, that the Framers gave Congress a method to re-check the president's veto power. (A two-thirds vote of both houses of Congress can override a presidential veto, allowing a bill to become law even over the president's objections.)
And a third example: the Supreme Court has the power of judicial review, meaning it can overturn laws it rules to be unconstitutional. Even if the president and Congress agree to pass and sign a piece of legislation into law, the Supremes can toss it out if they feel it runs afoul of the Constitution. Neither the president nor Congress has any power to override a Supreme Court decision, but the executive and legislative branches still do have important levers to check and balance the judiciary, mainly by influencing who gets to serve on the Court in the first place. Every Supreme Court justice has to be selected by the president, and the Senate also has to approve of the nomination; the idea of this system is to give the other branches a chance to prevent judicial tyrants from making it onto the bench in the first place.
The system of checks and balances sometimes results in dramatic clashes between the branches of government: Impeachment trials. Presidential vetoes. Supreme Court reversals of major laws passed by Congress. And so on.
But more often, the system instead encourages compromise. Executive officers, lawmakers, and judges typically prefer not to have their disagreements turn into all-or-nothing, win-or-lose showdowns with the other branches of government. Instead they try to avoid such confrontations by moderating their own policies before reaching such a breaking point. Often, for example, the mere threat of a presidential veto is enough to get Congress to alter a bill to make it acceptable to the president. Or the mere threat of Congress refusing to fund a military endeavor is enough to force the president to build up widespread public support before launching armed interventions abroad. Or the mere threat of Senate disapproval may be enough to prevent the president from nominating a controversial justice to the Supreme Court.
Americans often complain about the gridlock and inefficiency in their government. But that inefficiency is, to a certain extent, deliberately built into the system; it's the almost inevitable outcome of the system of checks and balances. The neverending system of compromises required by that system can be frustrating, at times, but it has also proven remarkably successful at preventing the rise of tyranny in America. As the old saying goes, Mussolini (the Italian fascist dictator of the 1930s) made the trains run on time; in other words, tyranny was efficient. By contrast, the American system of checks and balances may not make any trains run on time. (Ridden Amtrak lately?) But the system of checks and balances has prevented the rise of any American version of a Mussolini. Which was exactly the point.