In some ways it is harder to learn to sing backup that being a solo singer. When you are singing solo, you do not have to worry about blending in or following someone else, they have to follow you! Not so when you are the second singer or "backup" as it is known. When this is the case, you have to be aware of the main singers phrasing, nuances and even possible mistakes.
Listening is the most important skill a backup singer has to learn. It is far more important to blend in and not over shadow the main vocal than it is to showcase your own voice. This is not the time for standing out, you are there to enhance, color and accentuate the main vocal, not to over power it. You will not be asked back if it is felt you are only interested in your self. The best backup singer is one who goes virtually un-noticed. Sometimes people may not even remember that there was a backup singer, this is ok, and you did your job!
In In order to blend in with the main vocal there are many things you must keep in mind. First, position yourself where you can see the face and lips of the main vocalist. This is so very important. You cannot know when they are going to start a phrase or end one if you cannot see them. Also you will definitely not be able to react when they make a mistake. Sometimes you will need to sing a different verse for example because they started on the wrong one. If you are just looking at the music or going by what you have memorized you will be singing over the main vocal with a different verse and guess who will be blamed? Sometimes they will indicate by a nod of the head or a hand gesture that they want to repeat a section or go back to "the top" and you will not know this if you have your head buried in the music in front of you. Learn to watch them all the time.
This goes along with watching the main vocalist but also deserves special mention. You must ideally start the phrase with the main vocal and end the phrase with the main vocal. This is very difficult to do but there are some little tips to doing a good job even if you are not familiar with this particular vocalist or their phrasing habits. (And everyone has theirs) First as mentioned before, watch them very carefully to see when they actually start the phrase. Begin your line only when you are sure they have started and start yours quietly. This allows room to grow and to blend in imperceptibly. This seems like a very easy thing to do, but most people forget and try to jump in where it seems best and they end up sounding awful because they came in at the wrong time. If you start quietly and come up to the level of the main vocal you will have much more success in this regard.
The same thing applies in reverse in the closing part of the phrase, the end if you will. Start preparing your ending before they get there. Anticipate the end coming up and watch them. You will know by observing when the end is coming up. Make sure you watch their lips and face to see when they might end. Do not get caught singing after they have ended! This can be very, very embarrassing. Also a note about beginning and ending consonants. Sometimes when two or more people are singing a phrase there can be multiple popping "p's" or sibilant sounds as each singer attempts to sing at the same time. A neat trick to avoid this is to drop the first consonant entirely when you are the backup singer. If the line is something like "peter piper picked a peck of", you as a backup vocalist might sing "eter-iper-icked-a-eck-of". This enables the main singer to have the definite first consonant of the phrase without you running over them or being out-of-sync with them. So when you are called to be a backup singer, remember you are there to blend in, not stand out. Enjoy!