As I've been following this thread, a couple of things confused me:
How are the player contracts structured in the European leagues - is the contract with the team or the league?
The team.
What is the "transfer window"? Is it like free agency in US sports where the player can hold out for the best offer?
Globally, players contracted to other clubs can only move in the months of July, August and January. Players and clubs can negotiate outside of these times but the moves are not official and the player cannot represent his new club until the next window.
A player is contracted to his club who hold his playing registration. If another club wants to sign that player while he is under contract then they are not allowed to talk to the player or his representatives until being given permission to do so by his club. This permission will normally be granted when the two clubs have agreed compensation in principle - this is known as the transfer fee. (When, for instance, you hear that Kaka has signed for Real Madrid for £56m, the £56m is his transfer fee - the current world record)
There are exceptions to this for players under the age of 17, under the age of 23 and over the age of 30. Players older than 23 may unilaterally buy out their contracts but the rule about not talking to other clubs still applies so they must do so out of their own pocket.
Once a player is within the last six months of his contract, he may speak freely to other clubs.
An out of contract player may move clubs at any time and is not restricted by the transfer window. If the player under the age of 23 who is out of contract moves, then the club that developed that player will receive a compensation fee to be decided by an independent tribunal.
Clubs may also loan players to other clubs in exchange for the receiving club paying a portion of the players' wages, a loan fee or both. Typically, players on loan are not allowed to play against the club they are contracted to. Normally, loans are only allowed to be arranged within the appropriate transfer window with the exception of emergency loans where clubs are given special dispensation to sign a player, normally restricted to goalkeepers, on loan when their normal goalkeepers are injured.
It should be noted that the English Coca-Cola Football League (the three divisions below the Premier League) allows transfers more flexibly. English football's lower leagues are a bit of a special case since they are highly professional teams with well paid players employed by clubs who often find themselves needing the money from transfer fees or just needing to get players off their books. Transfer fees are becoming increasingly rare in the lower leagues.
The only league I am aware of that operates a centralised contract system is the MLS, and I believe FIFA are only turning a blind eye to that because America is the last big market they haven't really tapped in to yet.