One thing that also intrigues me is how the draft works from the players perspectives - do they for instance sign a contract beforehand that says they will go to whatever team choses them or can they say no entirely and go anywhere? It has always seemed a bit bizarre.
Most players are usually pretty realistic about not being able to play for the teams they supported while growing up. Usually you won't have a player complaining about being drafted to a team, even though it's a shitty one. It's far better for their public image to claim that they're looking forward to the challenge of helping the team get better. What happens is if they cannot agree to a contract with the team they can refuse to sign and by the time the next year comes around, they reenter the draft, only this time the team that drafted them last time cannot select them.
The drawback to that is that this gives the player (and his agent) a horrible reputation. This also means they haven't played for a full year so they'll be viewed as either out of shape or maybe not as sharp as they would be coming out of school. As a result they're probably going to be drafted a lot later than before which usually means missing out on A LOT of money.
Usually this never happens because of that. The money is too important. Usually the team will find a way to trade the player or they'll come to some sort of agreement. The last time a player reentered the draft, it was because he was drafted in the baseball draft as well and he decided to play baseball, only his career flopped so he went back to football the next year.
Every so often though you do get a potential #1 pick stating that he won't sign a contract with the team that picks first in the draft, usually because they don't believe that team is committed to winning (keep in mind the #1 pick goes to the worst team in the league which may be there because they were unlucky or may be there because they're a shit organization). Then that team can either draft someone else or draft the guy anyways and trade him (as was the case with Eli Manning).
Thing is though the players that aren't drafted actually become available for signing (like a free transfer) so a lot of players that might not necessarily be highly rated do get a chance to try out for their local teams. Unfortunately most of them don't make it to the 53 man roster, but some of them are lucky enough or talented enough to force their way in.
So to answer your question, a player's entry into the league must be done through the draft, I believe. If they're drafted, the team that drafts them owns their rights for a year. If not, then they're free to go whereever.
This relates to what you asked about the hierarchy and a little bit to Alex's response. With the FA you have a bunch of separate clubs that enter into agreement to play matches with each other under specific rules. Essentially though they're all separate clubs (I'd say businesses but Europeans tend to bristle when we use terms like "franchise"). The NFL is far more collective. They did start the same way, but once they formed the league, the league itself became an enormous entity that pretty much controls everything, including the players that come in and out of the league. I'd equate it to FIFA or UEFA a little bit, but I'd say they have a bit more control, though with regulations like financial fair play, the FAs are getting a bit stronger.
