All I want to do is change the backgroundColor of fiveMinButton and tenMinButton when fiveMinButton is clicked. Why doesn\'t this code work? @IBAction          
        
You are writing the action method inside the viewDidload, Try this:
override func viewDidLoad() {
    super.viewDidLoad()
    fiveMinButton.addTarget(self, action: #selector(action(_:)), for: UIControlEvents.touchUpInside)
}
func action(sender: UIButton){
        if sender == fiveMinButton {
            fiveMinButton.backgroundColor = UIColor.gray
            tenMinButton.backgroundColor = UIColor.lightgray
        }
}
There is 2 problems with your code.
selectoris wrongviewDidLoad that will also wrong, so write that method outside the viewDidLoad as class method.  For first one change selector like this.
fiveMinButton.addTarget(self, action: #selector(action(sender:)), for: UIControlEvents.touchUpInside)
For second problem just write action out side the viewDidLoad.
If button type rounded rect ..Color cannot be applied. So set the button to type custom and good to go
self.myButton.backgroundColor = UIColor.redColor()
or you can set the background image or image to the button to gain the color
self.myButton.setBackgroundImage(UIImage.imageNamed("myButtonImage.png"), forState: UIControlStateNormal)
or
self.myButton.setImage(UIImage.imageNamed("myButtonImage.png"), forState: UIControlStateNormal)