Learn the key and lead, or main melody, of the song you wish to sing background to and know it well. Harmonies use other notes in the same key to shape themselves around the main vocal part.
Play this lead on your instrument, preferably a piano. While playing, pay attention to the main key the song is in, as well as the keys that are used throughout the song. Beginner harmonies with lead and background vocals are structured much like musical major and minor chords, or the first, third and fifth note in a key.
Incorporate harmonies on the instrument. If your melody is based on the home note, or first note in a key, the first harmony will be based on the third note. Thus, if you play C, G, F for your lead melody, play E, B, A for your first harmony. First harmonies almost always move exactly how the melody moves because they are the harmonies closest to the melody. Thus, in the key of C, if your melody is based on the third, or E, your first harmony will still be based on the C. Second harmonies involve fifth notes or harmonies that are farther apart, and do not always change when other notes do in order to maintain the correct chord structure.
Practice singing along with these harmonies. Also, exercise your ear for harmony by playing various chord progressions and singing only the first, third or fifth note of each chord.
Listen to music as often as you can and stop singing along to the melodies. Hear what harmonies are sung on your favorite songs and practice singing only them. Part of learning background harmony is to develop an ear for it, or make it instinctive, so that you can sing harmonies to songs you have never even heard before.
Get together with another singer who knows harmony, preferably the person you will sing background to. Sing the melody of a song and let the other person show you one or both harmonies. Switch parts, and let the singer give you advice on how to be a stronger harmonizer. This, combined with practicing to every song you hear, is the most powerful way to learn harmony, especially if you do not play an instrument or even understand the theory of how you are harmonizing.
Play simple songs when practicing harmonies. In other words, play songs that have the same lead vocal structure and the same three or four chords throughout their duration.
Practice confidently and louder than you would when performing. When you are singing to a recording, you can really let loose and test your harmonies to see if they fit well and consistently with the song.
Choose the harmonies closest to the melody if you are the only background singer. Second melodies are harder to process while singing and can often become single, drone notes if you are not careful.
Sing less forcefully than the lead singer. Unless you are singing a duet, your microphone should be turned down and extra reverb may even be added to your voice.
Avoid vibrato as much as possible. Your vibrato may not match that of the lead singer or other harmonizers, and this can easily ruin the sound of a performance unless it is done extremely well.
Blend with other singers. The intent of background vocals is to enhance the performance while still emphasizing lead singers.
Sing with confidence. Know your part well. It is often hard to hear yourself and other singers as well as you would like to, and you are likely to miss notes or become completely inaudible if you lack confidence. Plus, if you do not know your part well, you may jump into harmonies that are supposed to be sung by other vocalists.