Diatonic refers to using the notes available in only a single, specific scale and not to my beverage of choice (and you pretty much have to either be a New Englander or at least know one for that to not go totally over your head, so don't worry!).
As mentioned, there are no flats or sharps in the C major scale. The notes are C, D, E, F, G, A and B. These are the notes that are diatonic to the key of C major. Bb or F#, not being part of the C major scale, are not diatonic to the key.
I'm sure this is going overboard to prove a point, but notice that none of the diatonic chords in the key of C major contain any flats or sharps. This makes total sense because none of the notes in this key are either flats or sharps.
The reason for doing this is to give you a "generic template" that you can then use to help you study the chord progression, transpose the song into a different key, determine which scale might be best to use for soloing, or any number of other helpful musical tricks.
If you take a moment to look at this template, and then take another moment to understand that it works for all major keys, then you can make some very important observations. When using the diatonic chords of any major key, the "I," "IV" and "V" chords will always be major chords while the "ii," "iii" and "vi" chords will always be minor chord. To make matters even simpler, most players don't need to worry about the "vii" chord, so just take it out of the picture altogether for now!
Suppose we had used the key of A as an example. Well, we could go and write out the entire A major scale and then figure out what each of the diatonic chords might happen to be. But if you already know that the A major scale is A, B, C#, D, E, F# and G# then you can easily make the connection that the diatonic chords in the key of A major are A, Bm, C#m, D, E and F#m (and yes, you can certainly include G#dim in that list but again most of you aren't going to ever find yourselves worried with that).
Let's look at a practical example of how this information can help you. Say you sit in on a jam with some other musicians and someone says, "Let's do blues in the key of A." Well, having read our "microlesson" on the twelve bar blues format, you'd know that blues songs typically involve just the I, IV and V chords and you'd also know that, in the key of A major, those chords would be A (the "I"), D (the "IV") and E (the "V"). Even if you'd never heard the song before, you could jump right in and take part in the jam without worrying about not knowing the chords.
In fact, whenever faced with any song you've not played before, the diatonic chords will give you a fairly good idea of what chords are likely to show up in the song. For songs in major keys, the I, IV, and V chords are typically the ones you're most likely to encounter, with the vi, ii and iii chords not all that far behind.
The more songs you know and play, the more chord progressions you run into. This experience with chords will eventually translate into knowledge of what chords tend to follow one another. In the Guitar Noise article called A Before E, we give you a bit of a head start with this information. Of course, nothing is ever written into stone when it comes to music. There will always be chord progressions that break out of the standard musical packages that we all know. But the knowledge of the diatonic chords in the key of the song you're in can only help you in the long run.