There are a few things to analyze here in determining the correction procedure/product.

Are the eyes puffy? If so, use a skin cream that contains caffeine like MAC’s Fast Response Eye Cream and anything cool and calming like a refrigerated eye mask or cucumbers.

Are the dark circles predominantly blue/green/grey, purple, or yellow? For blue/green/grey use a product that has a little pink in to brighten up the area. The best product I’ve used for this type is Givenchy’s Mister Light, which is a liquid-ish concealer and brightener in one. For the yellow, sallow eye circles that can occur in olive tones like some Latinos or Asians, use a concealer, or primer like Smashbox’s Photo Finish Primer in Color Balance that has a bit of lavender. Some recommend using something with a yellow base like for the blue/green/grey but really, that’s best for purple under-eye shadows, which I actually find to be the least common. Here’s a good color wheel chart to explain. Mister light as well as Makeup Forever’s Lift Concealer come with different bases, some yellow and some pink. 

Start the application with a single, fairly thin, layer and add additional layers as needed. It’s ok to layer with more than one color correcting shade using something like Makeup Forever’s Camouflage Cream Palette. For this start with the rosier or more lavender shade and finish with the more yellow shade if needed. There will be a certain point that’s as good as it’s going to get…. don’t push it. Take a fluffy eye shadow brush with a little translucent powder to set the coverage and make it last longer. And don’t be too shy to freshen up those babies later in the day.