Chords are combinations of two or more notes played together. Commonly chords will combine the 'root' note, the fifth note in the 'root' scale, and often the third note of the 'root' scale as well.
By playing more than one note at once the artists can provide not only a melody but harmonious voicings that create a mood and add layers of sound to what could be only a one-note-at-a-time melody.
Chords can also help convey emotional meaning. For instance, major chords have a happy, up-beat sound, where as minor chords have a sad or melencholy feeling to them.
Interestingly, the human ear can descern certain information about chords. By simply playing the root and the fifth note of the 'root' scale the ear can determine which note is which (root or fifth).
Chords allow musicians the ability to convey a lot of meaning with as little effort as possible. The entire chord does not even have to be played at the same time. Appregios move through the notes of a chord one note at a time, but still play each of the notes of the chord and no other notes.
Most songs also include bass lines, which will generally follow the root note of each chord and often the fifth or third notes as well. Most songs also include lead instruments, such as guitars, violins, mandolins, saxaphones, etc. that play one-by-one notes or occasionally multi-note parts.
The chords played provide a landing-pad of sorts for lead and accompaniment instruments to take off from. The rhythm instruments, or the instruments that play the chords (often acoustic guitar or piano) help to move the song along, providing the underlying core for the entire song.
That's what they mean when they say they put the right chords together.
One good example is a very common song that most people have heard at least once or twice, "Sweet Home Alabama". This song is built of three chords, D major, C major, and G major. The little 'riffs', or one-note-at-a-time guitar parts between chords, are based on these three chords, and the 'root' note for the whole song is G. Every note in the song, even the wild guitar solos, are drawn from the key of G, specifically G major and G melodic minor.