In most countries, seasons start in October and end in April, more or less....but in most leagues there is a determined date till when you can register players in order that they can be elegible to play in the season. Because of that, most leagues open in January the market again, so the clubs can sign with new players. Its the winter market.
Players usually sign a contract for the hole season, but there are exceptions, the contracts are made by time, so some players only signed till December, others the contract only starts in January, it is really open in that way. And sometimes, clubs and players decide to terminate contracts, if it is for poor performance, or if its an injury or whatever reason that is, if its a mutual agreement that is no problem. Sometimes there are cases involving fightsin breaking this contract, in both sides. If a clubs stop paying salary, or just if the player leaves the cub without any permisson from the club, that stuff...but over all that is rare, most contracts are terminated in mutual agreement I would say.
The league in China usually ends before most leagues in Europe, that is why you see some players signing other clubs there, after they end their season there. In Puerto Rico in the other hand, the season only starts in January, so you see a lot of players that have contracts with clubs there, playing in Europe before, between October to December, since they are free during that time.
So it can change depending the country but for the most part, Its one season per year, from October to April.