When you sing in tune together with other singers, you are creating the sound of an instrument playing a harmonic interval or chord. In a group setting, each harmony singer usually sings a tone that is part of the chord being played by the other instruments in the group. As with most other aspects of music, you will benefit greatly by understanding what you are doing - in particular how to construct chords.
You will always have a much easier time with vocal harmonies if you understand the chords in a song and are able to sing the tones of those chords. The simplest form of harmony singing is when you create a second vocal part that goes well with the lead singer's main melody. You often hear this in the refrains of popular songs, where the second voice 'adds weight' or gives emphasis to the chorus of the song. The two tones sung simultaneously by the lead singer and harmony singer create a harmonic interval.
When learning to sing harmony, try listening to some of your favorite songs that have two-part vocal harmonies and see if you can hear the intervals created by the two singers. Thirds and fifths are particularly common. See also if you can work out which chord degree is being sung by each singer.
Our game Vocal Match will give you great practice in singing parts of common harmonic intervals (levels 6-10) and chords (levels 11-20).