To remove a member from a board, click the avatar in the members section in the sidebar, and select "Remove from Board". You must be an admin to remove a member from a board.
When a member is removed from a board, the member will be removed from all cards on the board. This simply means they will no longer be a member of those cards. However, any other actions on the board (e.g. comments, checking of an item on a checklist, etc.) will remain. For example, if the member left comments on the back of the card, the comments will remain even if the member is removed from the board.
See also Removing people from an organization.