The transfer market
How the market works: every player is owned once, and moves through auctions or direct offers.
Every player in a league exists exactly once. At any moment they belong to a club, or sit on the open market as a free agent, never both, and never neither. A signing doesn't create or destroy anyone; it just moves them from one owner to the next.
Two ways to move a player
Free agents, and any player a club chooses to list, go to auction, where the whole league can bid. A player still under contract who hasn't been listed can only move through a direct offer, negotiated privately between the two clubs that hold and want the player.
The market stays stocked
The system keeps a run of free agents up for auction at all times, so there's always something live to bid on even if nobody has listed a player recently. A club can also put one of its own players up for auction, up to a small per-club cap, rather than waiting for a private offer to land.
Next
See Auctions and proxy bids for how bidding works, Direct offers for private deals, and Market value for what a fair price looks like.