A photographic Grey Card can help you find the correct exposure when shooting in manual because your camera thinks the world is - you guessed it - grey! You can also use photographic grey cards to help with colour accuracy.
Why do cameras think the world is grey? A camera doesn't know how much light is available in the same way an ambient light meter does. A camera can only assess how much light is being reflected by whatever you point it at.
Some things are more reflective than others and to make up for this manufacturers have programmed the light meters to equal everything out to mid grey. And because not everything is equal to mis grey your camera can sometimes get the wrong exposure. When you know how to use a Grey card you can make sure your exposure is correct.
For colour correction a neutral Grey shows up colour casts. For example if the light's a bit blue the Grey card will look a bit blue - or pink or green or whatever. And because you know the Grey card has no colour of it's own it's easy to adjust the colours in post production - especially handy if you're shooting RAW files.