Start with a major scale. For illustration, we'll use the key of C. Write the notes of the major scale with lower-case letters and number them with Arabic numerals:

   1 2 3 4 5 6 7
   c d e f g a b

These numbers can be used to list the notes in a chord, for example a C major chord has notes 1, 2, & 3, which are c, e, & g.

When talking about chords, we use Roman numerals and capital (upper-case) letters:

   I  II  III  IV  V  VI  VII
   C  D   E    F   G  A   B  

In the key of C, the chords you mentioned are the I, IV, & V chords. Jillions of songs use just these three chords, or sometimes even just two of them (I & IV or I & V). When I talk about very broad and general chord patterns, I use just the numbers. When talking about more narrow chord patterns or the chord sequence for a particular song, I get more specific, as follows:

The major (tonic) chord uses just the number: I=C major, II=D major, VI=A major, etc. Other chords are noted just as if you were using the letter name of the chord: I7=C seventh, IIm=D minor, IIIm7=E minor seventh, IVmaj7=F major seventh, etc. Likewise flats and sharps: IIIb=E flat major, V#dim=G sharp dimished.

As you can see, these numbers have nothing to do with time signatures or describing a chord shape or variation (in IIm7, for example, it's the "m" and "7" that give you the specifics, not the "II"). So what's their value? Chord patterns and transposing.

Any song's chord sequence can be written out in the Roman numeral form. You will start to notice many different songs, regardless of the particular key you usually do them in, have very similar or even identical chord patterns. If you train your mind to think in these Roman numerals, transposing to a different key is much simplified. For an example, we'll look at "Happy Birthday":

  In the key of C:
        C           G          G           C  
  Happy birthday to you, happy birthday to you
        C             F          C        G  C  
  Happy birthday dear you, happy birthday to you
  
  With Roman numerals:
        I           V          V           I  
  Happy birthday to you, happy birthday to you
        I             IV         I        V  I  
  Happy birthday dear you, happy birthday to you

Now let's look at a couple of other keys:

  Key of G:                    Key of D:
   I II III IV V VI VII         I II III IV V VI VII
   G A  B   C  D E  F#          D E  F#  G  A B  C# 

You can make a similar chart for any key, based on its major scale. Pick the key you want to do the song in, stick that key's chords in where the Roman numerals are, and you're ready to go. To do "Happy Birthday" in G:

        G(=I)       D(=V)      D(=V)       G(=I)
  Happy birthday to you, happy birthday to you
        G(=I)         C(=IV)     G(=I)    D(=V) G(=I)
  Happy birthday dear you, happy birthday to    you

Try writing the numbers on any song you do or are trying to learn, and after a while I believe you'll see how it simplifies many things for you. Good luck!

(P.S.--feel free to post here or send me a PM if there's anything I can clarify for you or help you with.)