The first day of June is an important day when it comes to the NFL calendar. Before today, if a player was released, the club that released him would be on the hook for any remaining signing bonus and/or guaranteed money in one lump sum. That could effectively wreck a team's salary cap. As of today, however, teams may begin releasing players and spread the salary cap hit over this season and next. That savings usually leads to some veteran free agents suddenly becoming available.
Of note, the NFL also now has a "June 1 cut" designation, in which two players from each team who are released prior to the first of June can have their remaining dead money count over the two seasons. This desigantion was designed to allow teams to still receive the benefit of the June 1 cut, while allowing the player to hit the market earlier in the offseason, giving him a better chance to find a team with a need. This has reduced the impact of the actual June 1 mark, but there should still be some veterans becoming available over the next few days.
The Miami Dolphins used both authorized June 1 designations in the 2013 offseason, releasing linebackers Karlos Dansby and Kevin Burnett at the start of free agency. The club did not use it this offseason.
Fox Sports' Alex Marvez pointed out that there is also another June related step in the free agency process. According to Marez, compensatory draft picks are only calculated prior to the calendar moving to June 2. That means, after filling major needs early in free agency, teams could wait to see what June 1 cuts are made, before signing additonal free agents starting on June 2, if they believe they will receive quality compensatory picks in the 2015 NFL Draft next offseason.
Compensatory picks are awarded to teams who lose more (expired-contact) free agents than free agents they sign in a given offseason. the league annually hands out 32 of these additional draft selections, spread over rounds three through seven. A team who has lost multiple players through free agency could wait until June 2 to sign another free agent in an effort to assure they receive a share of the compensatory picks, or to ensure the picks they do receive are earlier in the draft.
Whether it's the waiting for salary cap relief by not releasing a player until June 1, or if it's wanting a compensatory pick so a team waits until June 2 to sign someone else, June is an important stepping stone in the offseason process for the NFL, effectively creating a second round of free agency.