Adding people to a board
To add members to a board, select "Add Members..." from the board sidebar. Search for a user by name or enter an email address to invite them to the board. Click their name to add them to the board. Depending on the board settings, you may need to be an admin to invite someone to the board.
You can add members to boards even if they are not already using Trello. Just enter the person's name and email address and click "Send." The user's permission level will be considered "Virtual" until they join Trello, but you can still add the user to cards.
Allow Org Members to Join
If your board is a part of an organization and you would like any of your organization's members to be able to join the board and collaborate, you can select "Allow Org Members to Join" from "Additional Settings" in the board menu. When you select this option a checkmark will appear next to "Allow Org Members to Join" and your organization's members can add themselves to the board.
You can further control board permissions in the "Additional Settings" option in the board menu. Under "Commenting Permissions," you can select who is allowed to comment on the board: no one, just board members, board members and observers, all organization members (whether or not they're board members), or any Trello user.
Using the "Invitations Permissions" menu, you can select who's allowed to invite new members to your board. If "Admins" is selected, only board administrators can invite new members. If "All Members" is selected, any board member can invite a new member to your board.