Diatonic is like playing only the white keys on a piano. It means you are sticking to a specific set of notes in a certain key, and only using the notes that belong to that key. It's like coloring with only a certain set of crayons that match the colors in a picture you're trying to color.
Chromatic is like using all the colors in a big box of crayons. It means you are using every note available, including the sharps and flats that don't belong in a specific key. It's like coloring a picture however you want, even if the colors don't match the picture perfectly.
So, when someone is playing diatonically, they are sticking to a specific set of notes that fit together nicely in a key. When someone is playing chromatically, they are using all of the available notes, even if they don't perfectly match the key they are playing in.