If you use the grid rules in the DMG, you center the circle on a grid intersection and any square that's at least half covered by the circle is affected by the spell. Non-circle areas affect any squares they overlap, regardless of how much is covered.
Xanathar's Guide to Everything has two other methods you can use. One of them involves making cutouts of different spell shapes and just overlaying them on the grid; whatever squares it overlaps are affected. This is basically how spells work without a grid; you're just using a visual aid.
The other method involves placing dice or other tokens on the grid to fill out the area.