Each team is allowed to designate one of their players who is going to become a unrestricted free agent with the 'franchise tag'. An unrestricted free agent is a player whose contract has expired and is free to negotiate with any team without his present team having the opportunity to match the contract negotiated between the player and the other team. There are two types of franchise tag ... exclusive and non-exclusive: Exclusive: The player must be offered a one year contract 1) at a salary of the average of the top five salaried players at his position, or 2) at a salary at least 120% of his previous year's salary, whichever of the two is the higher amount. The player may not negotiate with any other teams even though he is a free agent. Non-exclusive. Same as the Exclusive player except the player is allowed to neogtiate with other teams. Should the player agree in principle to a contract with another team, his present team is allowed to match the contract and keep his services. If his present team does not match the contract, the player becomes a member of the other team and the other team must 'pay' two first round draft choices as compensation to the player's now former team. The exclusive tag is a way for the owners to get around free agency. The exclusive tag does not allow the player to negotiate with other teams and the non-exclusive tag states the team will be compensated if the player signs with another team.
According to this news story, players can be traded after having the franchise tag applied to them: http://espn.go.com/nfl/story/_/id/7569946/philadelphia-eagles-likely-franchise-wr-desean-jackson-report-says
Once they have signed a contract or the player received a franchise tag
A Franchise tag is not a German phrase. It refers to privileged labeling, i.e. in professional sports. Also see weblink below.
If a player is traded, the franchise tag is basically null. The franchise tag applies only to free agency. However, if a team went through the effort and expense of putting the franchise tag on a player (paying that player the average of the top 5 salaries at the player's position), the possibility of them trading that player is highly unlikely.
A franchise record is a record that is just on a team not in the NFL
New Mexico does not have an NFL franchise.