Why would a '84 Mercury Cougar make a clicking sound when you try to start it and nothing else?

Battery terminals are corroded no doubt... It could be the battery terminals, especially if all other electrical devices such as the dome light and dashboard indicator lights turn off when the starter clicks. If all other lights remain on but are dim, it could be a low battery. If everything stays on and the brightness of the lights doesn't really change when you try to start it, it could be a bad starter. If it's the battery terminals, clean them and use a terminal protectant to keep the terminal from corroding again. But recognize that when a battery is starting to go bad it often starts giving off acid fumes that start or accellerate the terminal corrosion process. After cleaning the terminals, take the car to one of the do-it-yourself auto parts stores and have the electrical system checked. If the battery passes, keep watching it for potential problems.