Color correction is the technique of changing a color into another color, and color grading is actually the same things.
From what I learned, color correction is mostly referred to white balancing, the technique of correcting the white in your video with the real white we saw in real life, when light go inside the camera’s lens it gets easily blended with all the artificial lights around it, so for example if you record in some interior with yellow lights, the main tone will be yellowish in the final video, not only artificial lights but also the sun light can change the white color perceived by the camera sensor. The white tint can be fixed during recording at some degrees, using the white balance function integrated in your camera but if you keep the white balance at a fixed preset, it should be corrected later in post with an application that have color grading tools.
On the other hand color grading is mostly referred to the technique of changing a color to another color, there is no rules or limit when color grading, the color grading change the atmosphere of your film and the feeling and emotions the audience has when it see it, it is only up to you how you want your film looks like, there is not right or wrong way of doing it.
It goes without saying that the white balancing should be done before color grading. This is mostly true when you use LUTs, since the LUT preset is usually created starting from a well exposed and well color corrected image, if you apply the preset without white balancing the result might be different from the expected one.
What is a LUT?
A LUT, or ‘Look Up Table‘, contains a set of numbers which are looked up by the software or hardware you are using in order to deliberately change the colors of an image.
A LUT can be used for technical purpose, for example to convert a LOG footage from a DSLR into a more common format, so the preview on the camera lcd will still look contrasted and colorful during the record of a LOG footage. LUT is also used for creative purpose, to alter the image, saturation, contrast, hue level, and much more, to make your film look more cinematic or just better looking, if you don’t like the ‘cinematic’ word.