There are three different types of icons that could show up in the departments. The exclamation mark with a staff icon appears when a department isn't functional because there aren't any staff members assigned to it, or because the staff members assigned to it are out of stamina. Pressing this alert will open the Management screen where you can switch staff members.
The big, red exclamation mark appears when a department isn't functional because it needs something to start or end a cycle. For example, if the reception area doesn't have space for more patients, or the operating room doesn't have patients to treat and/or enough beds for treated patients to rest in. Pressing this alert displays more information about the issue.
The exclamation mark with a bed appears only in the ward. It appears when all the beds in the ward are being used, which prevents further patients from being admitted to this department. Pressing this alert displays information about the beds in the ward, and in here you can free up beds, which will cost you some Golden Hearts.