Our eye area is very sensitive and needs hydration. Yes, your daily face moisturizer will do it, but there are eye creams that are designed for the eye area. I use two at the moment, and anti-aging eye cream(Line Reducing Eye Brightening Concentrate) and a more hydrating eye cream(Kiehls Creamy Eye Treatment with Avocado) for the daytime. I also overlap the moisturizing eye cream over the anti-aging eye cream since it can be a bit drying underneath the eye area. Ideally you should be applying eye cream both in the daytime and nigh-time. Make sure you use your ring finger since the eye area is delicate and the skin is thin as well. In my photos down below you will see what is ideally for the daytime application and the nigh-time application. There is also a correct way to apply eye cream and incorrect way to apply it.
Eye cream tends to travel upwards when you're sleeping and that is why you should apply it further down to prevent from causing even worse puffy eyes. If you have been using eye creams and still suffer from puffy eyes or dark circles it might just the technique you're using. Change it up and start seeing the difference.