Gonna weigh in here very quickly (I have to run soon) -- a lot has been covered.
One thing I want to point out re the US Supreme Court is that it is a common misconception that anything and everything can end up there. "I'll take it all the way to the Supreme Court!" is a mistaken rallying cry when the truth is that many, many cases don't qualify because there is no federal jurisdiction.
The US has state courts and federal courts (there are also maritime courts, tax courts, etc. but this is meant to be an easy explanation). State courts handle state law, e. g. most criminal cases, trips and falls, dog bites, etc. Federal courts handle federal cases. It's also a misconception that a "federal case" is a big deal. Well, sometimes it's not. A $25 employment dispute where there was a civil rights violation is federal, even though it is small potatoes.
Generally speaking, applications of federal law include:
- civil rights
- interstate matters, such as interstate commerce
- maritime matter (usually; this doesn't include local riparian rights -- that's the right to water from a river or stream, in general)
- international matters
- matters having to do with federal government
Federal law also kicks in when a matter has happened that involves more than one state and a certain financial limit has been reached. It was $10,000 when I was in Law School, then it was $25,000, and I believe it has been increased again. Think you'll never, ever get involved in a matter involving more than one state? Think again -- you might drive to another state and get into an auto accident, or trip and fall in front of a store on vacation, or be subject to noise pollution from a company incorporated out of state.
I have oversimplified this in a big way, and I realize the original question was more about the legislative area of US government, but the judiciary is equally important, as is the executive branch (the Presidency).