You asked how much power each branch has. There is a system of checks and balances, but it would be naive to think that each branch is equal.
The Supreme Court's word on an issue is final. It's the highest court in the land, and there isn't anyone with the authority to overrule it (the individual justices are another matter). The only thing is that there aren't that many things that the Supreme Court has a say on. The court cases that it rules on rarely have a big impact on national politics. It's most important function, deciding whether laws are constitutional or not, is rarely needed since Congress almost never passes any laws that aren't constitutional anymore.
The president is in charge of enforcing the laws (FBI) and conducting foreign affairs, peaceful or otherwise. He can veto any bill passed by Congress and nominate Supreme Court justices. Officially, that's the extent of his power but, in reality, he's much more powerful than that. The president represents the nation and, more importantly, his party. Therefore, he usually has his party's full support in all his actions so, assuming his party has the majority in Congress (as is the case now), they will go along with everything he does.
Congress, which predates the other two branches and is 100% democratically elected, has the most power...on paper, at least. They can pass laws, levy taxes, and declare war. They can even overide a presidential veto if they have a large enough majority that supports the bill. They also have the power to impeach the president, vice president, any member of the cabinet, and (I think) Supreme Court justices. Important note: public officials can only be impeached for treason, bribery, and committing a high crime or misdemeanor, not for being incompetent. The problem is that even when the president doesn't have a favorable majority in Congress, the legislature is usually still too split to overide a presidential veto, and the right to declare war is pretty meaningless when you have no control of the military.
Basically, the branches are equal in theory, but, in reality, the president is the most powerful figure in the country.