If you think gray hair or wrinkles are what makes you look old, actually the dark circles under your eyes can make you look more tired and unhealthy, and much older than you think!
While insufficient sleep and fatigue due to stressful work can be the main reasons for dark under eye circles, you may also have dark eye circles even if you get plenty of sleep. This will be the case if you are afflicted with allergies, eczema or nasal congestion, or if you smoke or drink alcohol, or it just simply runs in your family. These are just some of the reasons.
If you ask me how to remove dark circles in general, I suppose the cheapest way is to DIY, ie self care methods at home, such as use cold compress (eg chilled teaspoons, cooled used teabags) on your eyes to reduce the dilation, and thus the discoloration of the under-eye blood vessels.
Most importantly, get extra sleep. And when you sleep, try to sleep on two or more pillows to elevate your head. This helps to prevent puffiness caused by fluid pooled in your lower eyelids.
If you have some money to spend, than you may ask your doctor to prescribe you with special creams that help to reduce discoloration or prevent under-eye circles. These creams usually contain vitamins, such as vitamin C and vitamin E.
Other than that, apply concealer to hide those dark circles. Choosing the right color of concealer is important here. Use a peach-colored concealer if the dark circles are bluish.
As the reason for my own dark under eye circle - less than 8, sometimes 7, hours of sleep every night! 
