Each team has an exclusive contract with their players, and no other team can get that player from another team unless they complete a trade. Usually, this means trading one or more players and/or one or more draft picks.
Once that contract is up, a player can be a free agent and can sign with whomever they want (some restrictions apply depending on the contract).
In the NFL, rookies (typically) who are entering from college enter into the NFL Draft. There are rounds where each team, based on its record from the previous system, gets to choose which player to have exclusive access to signing. The team then enters into a contract with that player for a specific amount of time and a specific amount of money (there is a "pay scale" now in place to make that process easier).
The team with the worst record picks first, then next worst, and so on. If teams have equally bad records, they have different tie breakers to figure out which team was worse and that team gets the better pick.
Teams don't always pick in each round if they have traded that pick to a different team or they've been punished for one reason or another (see: New England Patriots), and there are also compensatory picks if a team lost a big player in free agency the year before, but that's a bit more in depth.
There are seven rounds and then the draft is over. After that, all players without contracts are free agents (rookies are called UDFA, undrafted free agents) who can sign with whatever team they want.
