It really depends on the situation actually. Most of the time, Congress and the President work together which greatly expands the powers of both. Even when different parties are in each branch, they still tend to work together. For example, Presidents are given a fairly wide latitude in negotiating treaties with other nations. Usually Congress approves; however Congress can refuse to ratify some of those treaties and they occasionally have. So would you consider treaty making something the President can do or not? On one hand, we have SALT and NAFTA and a whole bunch of other treaties the President negotiated that prove it is. OTOH, we have Kyoto as a major example where the Presidents negotiation meant dick all.
The President is charged with law enforcement. He does so through cabinet appointments, which are subject to approval by Congress. Congress also has the power to pass laws limiting who can be appointed or laws to change how lower offices are filled (for example, as a direct result of JFK & RFK, family members can’t be appointed anymore). At lower levels, the President can fire people, but that often has a great political cost. You may remember Nixon’s ‘saturday night massacre’, where he quickly went through several Attorney Generals. Perfectly legal, but a large part of why Nixon resigned. Bush got some flack over firing attorneys as well, but little came of that. Obama has been accused of allowing his justice department to ignore voter fraud and that may or may not become a huge issue (I’m not arguing pro or con on that, just pointing out the situation exists). So do those limits mean law enforcement is within his authority or not?
And lets not forget the veto. That is an astounding amount of authority. Granted, it can be overcome, but even so, it’s quite a lot of authority. Often even the threat of using it will force Congress to give concessions on issues the President cares about. As an example of the power this gives a President, a disagreement between Clinton and Congress on the budget, which the President has no authority over since that’s all Congress, ended up in the debt ceiling not being raised, again this is Congress’s authority, and caused a shutdown of the government. All because Clinton had veto power.
Then there’s Commander in Chief. How much authority that really gives and what checks there are on it is rather a gray area. Congress usually respects this. The laws they do pass usually aren’t major encroachments on the Presidents authority. Even so, the President often argues the laws are unconstitutional, but he follows them anyway. IIRC there are laws that require the President to brief Congress occasionally on military issues, and the President does so while insisting it’s a courtesy rather than a requirement. The few times military issues have become a major political fights, Congress usually loses.
Anyway, the point is, it’s really hard to say what exactly the authority of the President has. Clearly he has a lot, but much of it is tactfully granted by Congress, or at least by Congress not actively countering it. Despite bitter partisanship, the President and Congress work together and don’t push that much on which wields the biggest stick on what issue.