Transfer of player from one team to other create several complication due to involvement of Agent and contract agreement now a days. I will write what I read on sports site or some other sources. We can not give you perfect detail about this. I will talk about only professional player.

How Transfer works?

Now a days most player do have agent to represent them. So suppose one team want to buy that player, first they need to talk to player's agent then agent will discuss with his player about the offer. Player can be signed only when transfer windows open. For most European leagues summer transfer windows open from 1st of July to 31st of August and winter transfer windows from 1st of January to 31st of January. This rule is not mandatory for free agent.

Why do players need agent?

If your perform regularly at high level, you will get attention from big team. These team want you to play for them. They will give you pay rise, bonus and other benefits. But if you get attention from many teams, talking to them takes lot of time, and your too much attention on these can affect your play.
If you are a good player, you not only attract big club but you also attract marketing company. They want you to advertise their product and they will give you the big Pay Cheque. Discussion of contract with these company is not only tiresome but also it will distract you from football.
That's why players hire agent to deal with these situation and they only play football and do what their agent says. Players agent get paid for that by doing this job.

Actual Transfer?

Suppose a player play for club A and he performed significantly well for them. He will get attention from many big club. They want him to play for their club. Suppose that club is B. They will contact his agent and his current club A . Player's agent will inform his player of this offer. Then player will say yes or no. If no then agent and A will tell that club no otherwise agent will discuss with A and they will decide they want to sell this player or not. Now B will negotiate the transfer fee with A and player's agent. Not only that players agent will discuss the basic salary, performance bonus, release clause, match playing bonus, image rights etc. After long negotiation with the players agent, player will terminate his contract at club A and will sign new contract at B. Agent will get his fees(Transfer fees of player) from club A whatever they agreed upon .

Release clause?

During the contract negotiation with the club, either club or player want that in his agreement, there should be a release clause of fixed amount(which may change yearly wise) such that if some other club want to get this player and player's current club do not want him to go, either player activate his own clause by paying that amount and becomes free agent or other team pays that much amount. If club still refuses to sell him, other club can sue player's current club.
For example,
Arsenal activated Suarez clause in 2013 by paying 40 million pound + 1 pound. Liverpool refuses to sell and Arsenal could have sue Liverpool for that.

Image rights?

This is most controversial part in player transfer. If you are signing player like Messi, C. Ronaldo or Neymar, they are not only world class player but also most marketable athlete. They bring you sponsors, will increase your shirt sell or other merchandise item. You will use them as front face in your app, your video or in your site. You can use them as front face during team kit release. It is good in financial terms for club. These are all comes in image right. When player sign contract, in that contract there is clear mention of players image rights. It may be 100 % or 90 % or 70 % or 30 % or may be 0%. Club want maximum image rights of the player. This is the stumbling block of transfer and very controversial. Sometimes player do not extend their contract because either they are not happy with the salary offered or mostly because of image rights.
In 2006 as part of his negotiations with the Spanish football club Real Madrid David Beckham demanded that they relinquish their 50% share of his image rights, thus giving him 100% of all proceeds from the commercial exploitation of his image. Real Madrid refused, leading to David Beckham leaving the said club. He later signed a five year contract worth £128 million with the US club LA Galaxy which also included full control of his personal image rights. The case of David Beckham is significant in that he is now able to control how and when his image is used especially in endorsements and sponsorship deals.

Co-ownership deal?

Co-ownership is a system whereby two football clubs own the contract of a player jointly, although the player is only registered to play for one club. It is not a universal system, but is used in some countries, including Italy, Argentina, Chile and Uruguay.
This type of deal differs from third-party ownership, in that in the latter, the player's contract is owned by a non-footballing entity, such as a management company. For example,
The co-ownership of Alessio Cerci by Torino and
Fiorentina proved fruitful for Cerci and Torino, who had paid €2.5m for the first 50%, and picked up the remainder of his contract after a successful year for just € 3.8 m.

Buy Back clause ?

B ig team insert buy back clause during selling their player. So if in future players performance level increase drastically they can buy him at that fees which is written on buy back clause. This fee increases year by year relatively. For example,
Bojan(Bacelona) has 13 million euro buyback clause when sold to Roma in 2011.

Profit on Selling?

Some times club before selling their player insert a clause such that if his new team in future thought of selling him to another club, they will get some percentage of transfer fees. For example, recently
Garay(Argentina)has been sold to Zenit from Benfica for around €15 million . The reported fee of €6 million is only Benfica's cut of the total transfer fee. The remainder will go to Real Madrid (who own 50 percent of his rights) and a third party.

Loan deal?

Not only player can get permanent transfer to some club but can also play for some club for temporary period.
A loan involves a particular player being allowed to temporarily play for a club other than the one he is currently contracted to. Loan deals may last from a few weeks to all season-long and can also be for a few seasons
Typical loan deal involves loan fees which can vary from 0 to two digit million pound. Player wages(salary) will either payed fully by team who loaned him or his owner team or can be divided between the two. Club send their youth player on loan to some team in same league or in other to get more playing time so they do get experience from them. Some times loan deal involving some clause like you can sign that player for a fee agreed between both the clubs. For example ,
Zlatan Ibrahimovic transferd to A C Milan in 2009 on loan from F C Barcelona. They later made it permanent buy paying the fees agreed between the two clubs.
Loan fees has also some clause like player can not play against his parent team and if his current team want him to play against his parent team then they need to pay some agreed amount.
For example,
Courtois(Belgium) was not allowed to play against Chelsea, if Atletico want him to play they should pay around 5 million euro. But UEFA interfere in it and warn Chelsea that they will give them punishment if they did not allow Courtois to play.

Free Agent/ Released player?

Player who are not bounded through contract by any club is termed as free agent. So if any club want to sign him, they need not pay anything until and unless his agent demands something. Players become free agent either they released by their former team with mutual agreement before their contract expire or their contract expire. Team who are interested in these player can sign them before the 6 month of their contract length. So suppose a player contract was going to expire in jun 2014 they can be signed in January transfer windows or just before it. Free agents do not have to be signed during the normal transfer window that is implemented in some countries' leagues.
If a player is released from their club when the transfer window is closed, they cannot sign for another team until the window reopens. A notable case of this being Sol Campbell who in September 2009 was released from Notts County, just after a month from signing on a free transfer. He signed for his former club Arsenal in January 2010, after spending a few months training with the team to maintain his fitness
I did my best to write all types of transfer. Transfer is big thing in football. It involves several controversies. Sometimes deal goes down due to involvement of third party. Sometimes player and club do not agree on contract agreement either over player bonus demand or club want to retain their most image rights. Sometimes agent mistake knowingly or unknowingly create problem for player. Current situation of Messi in tax case. His father and his former agent was handling all his financial decision. To know exactly what happen in football transfer you need to ask managers in big league, or club owners or football agents.

Long Brilliant...

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