Two small black areas have appeared on the right side of my screen. They remain through all applications. How do I get rid of them? Or is the laptop beginning to break down. Everything else working all right.

Unless they are soda drops dried on the outside of the screen... they are dead pixels and there is nothing you can do about it. Two really isn't bad... but if they start to multiply then worry.

I assume they are tiny spots.. not larger areas.

White or black stuck pixels usually indicate a dead pixel depending on the type of lcd. If the pixels are colored then they are just stuck. Even if they are black and white there are utilities that will attempt to fix them (google "fix stuck pixels"). Typically, if the number of pixels is less than .01 percent, manufacturers do not want to replace them even under warranty. If there is a known issue with your specific model and the dead pixels continue to increase, check with the manufacturer or their forum for any recall or replacement options. Sometimes, even after warranty a good company will help you to keep your business.