It can be hard to identify song chords, especially if several different instruments are involved and the song is complex. It can, however, be done. Slash, the guitarist for Guns N' Roses, learned how to play guitar simply by playing along to his favorite records. You can identify song chords by listening repeatedly to songs and applying any music theory you have.
Learn some music theory. This will be a huge help in identifying a song chord once you are able to discern a few distinct characteristics about what you're listening to. Learn about scales, harmony, keys, etc.
Get to know common song patterns. Many pop songs in the 1950s used the I-vi-IV-V progression and many blues songs use the I-IV-V progression. Knowing these patterns and scales can help you determine the chords in a particular song.
Listen to the song over and over again. This will help you get a feel for all the chords involved, the structure of the song and the different changes in "feel" at different parts of the song. Some songs, such as Nirvana's "Smells Like Teen Spirit," use the same chords repeatedly for the verse and chorus and simply change the vocal melody. Others, such as Beethoven's "Piano Sonata No. 8," change chords often and it will be much harder to pick them all out.
Stop the song at certain parts and try playing different chords until it sounds right. Turn off the music. Sing the melody of the song and play chords along to it until they feel right. Sometimes, more than one chord, not the one intended by the songwriter, will sound right over a particular section.
Determine the bass note and notes of the melody. If you have an instrument nearby, you can figure out the bass note simply by playing along until you find the note that sounds right. Do the same for the melody notes.
The bass note being played is often the root note of the chord you're looking for. For example, the three notes in a Gmajor chord are G, B and D, the root note being G. If you can make out the root note, this will often help you determine the corresponding chord. In "Smells Like Teen Spirit," bassist Krist Novoselic plays primarily the root notes to guitarist Kurt Cobain's corresponding chords. Determining Novoselic's notes would help you determine the chords of the song. Often a bassist will play beyond the root notes of a song. Don't worry, however, as the notes they are playing must still fit within the framework of an implied chord.
Once you know the root note and corresponding chord, try different combinations. If the note is A, try Amajor, A7, Aminor, etc. Play them over the chord you're looking for until they sound right.
Figuring out the bassline can help you determine what key the song is in, furthering your possible understanding of the chords involved. Once you know the key of the song, you can determine all of the possible chords that fit within that key. For example, in the key of E, the chords are E, F# minor,G#minor, A ,B, C#minor, and D# diminished.
Listen to the other instruments. Get a feel for the chords involved. If you know the chords of a verse to be major and its chorus sounds much darker, this can help you determine that the chorus chord is a minor. Elton John's song "I'm Still Standing" is an example of a song with a major verse and minor chorus.
There are computer programs you can buy that recognize pitch and can identify song chords for you.