Registered play-cricket users who are members of a club's site and team squads within that site can indicate their availability for fixtures for those teams, via their profiles ('edit account' under their name, and then the 'availability' tab) once they are logged in to the club's site. Club site administrators can also set availability for members in their database who are not registered play-cricket users, via the appropriate tab while editing their record in Admin>Members>Member Database.
Although the League Registered Player role overrides all squad roles in terms of viewing the players available for selection, the availability functionality is squad role driven. You therefore need to put a player into the relevant team squads in order for him to be able to indicate his availability.
Administrators can see which players are available for a given game, by clicking on the Team icon next to that game in Admin>Fixtures. They can then use the functionality on that page to select a team and print it for the notice board and/or publish it to their play-cricket site - the published details appear by clicking on the game in the Fixtures section and can only be viewed by people who are members of and logged in to your site, so the opposition cannot see them.
You can also notify the players by email - tick the 'email players' option and those players selected with an email address in their profile will receive a message notifying them of the details of the game and with links allowing them either to accept or decline the invitation. These decisions are then visible to the site's administrators