Board admin members are unable to add cards in that board apart from the creator of the board.

[Ashutosh Kumar]

On Trello, is it like the person adding a board can only add cards in that particular board. And none of the other admin member are getting the option to add a card there.

[Torben_trello]

Every board member can add/edit cards. It sounds as though that user is a member of the team that board is a part of, but not a member of the board in question. Once added the members should be able to edit cards on that board: http://help.trello.com/article/717-adding-people-to-a-board

To be clear, being a team member allows you to see boards and members associated with that team, but does not automatically give editing access to individual boards.