Watch the Home Coming online guitar lesson by Jeff McErlain from Essentials: Chord Tone Soloing
What we basically have here is what we call a I-IV-V chord progression. What does that mean? We are in the key of C, so that would be the first chord in the key, and the common way of notating it is with the I. The next chord is F, which of course is the fourth chord in the key of C, and logically G would be the fifth. There is our I-IV-V.
The Roman numerals work as a system of analysis which easily allows us to transpose those cords into any key, and this the way musicians speak to each other. It's a form of shorthand. The Nashville number system is a different thing that us New Yorkers never use. If you live in Nashville, it is essential. I digress, but if I wanted to tell the guys in my band the chords of the song, I might say it's just a I-IV-V in the key of G for example. The guys would automatically know that the chords would be G, C, and D. That is much quicker than actually saying the names of each chord.
This could expand into things like another common chord progression ,the I-VI-IV-V, or the ubiquitous II-V-I in jazz. Definitely take some time and start to analyze the chords of each song you play and how they relate to the key you're in. This is standard for semi-pro or pro musicians, and once you get it together you cannot imagine not ever having used it.