In modern Western music, Major and Minor triads (triad just means a three note chord) are the two most common types of chords (chord meaning 3 or more notes at the same time) and they make up the vast majority of all the chords you'll hear in Western genres like pop, rock, indie, etc.
Typically music will have a melody line (what you might think of as the 'tune' - it's often the line that's sung, or what you might hum), and this will be accompanied by chords, which form the 'harmony'. The structure of these chords, for example whether they are major or minor, helps dictate the emotional feeling of the music.
Major and Minor triads are made using their own specific formulas. Both start on a root note (below, this is 'X'), and have specific distances between each of the following notes ('Y' and 'Z').
Major Triads = X (+4 semitones) Y (+3 semitones) Z
Minor Triads = X (+3 semitones) Y (+4 semitones) Z
Most importantly many people develop emotional connections with specific chord types - many people feel minor chords sound 'sad', whilst major chords sound 'grander', or 'happier'. We can listen for these sounds to determine the quality of the chord.