Many people consider sight singing and music reading to be synonyms. But, more specifically, music reading usually refers to being able to play music on an instrument rather than sing it. Of course this requires a certain level of understanding of music theory, so that you know which notes to play.
I like to compare music to a foreign language. Which takes more skill, to be able to read a language with understanding, or to reproduce the sounds? Obviously reproducing the sounds is generally a whole lot easier than reading for understanding. And so it is with sight singing.
The idea of reading music typically encompasses a number of areas: knowing the names of the notes, knowing the clef signs, knowing what key a song is in, knowing which notes are sharp or flat, and knowing what keys, fingerings, or positions correspond to which notes. None of this is necessary to sight singing.
There is an area of overlap between music reading and sight singing, and this relates to rhythm. To do either, you have to be familiar with reading rhythms. But this is not usually that area that seems so mysterious to the beginning vocalist. Singing the correct pitches is what seems like magic. You see a dot on a page and you're supposed to know what pitch to sing. If you're an instrumentalist, you simply use the correct fingering, and out comes the note.
If you knew that a language only had twelve unique sounds, with half of them being used much more often than the others, it wouldn't seem so daunting to learn to make the sounds. That's the way it is with music; you don't need to know much about the meaning in order to make the sounds.
To begin learning to sight-sing, there are just a few concepts that you need to know. You need to be familiar with the sound of a major scale (e.g. the first phrase of Joy to the World or The First Noel) and the concept of a home note, also called the tonic, note number one of the scale. You also need to know that the alternate lines and spaces on the music staff represent consecutive notes of the scale. That's all the background you need; you don't need all those other details that instrumentalists depend on. What could be simpler?
Sight singing is basically a two-step process. First you determine which line or space represents the home note, referred to as Do (dough). Then you just get used to how the other notes of the scale sound in relation to Do. This is a skill that improves with practice, but it's a whole lot easier if you can associate familiar songs with the various notes. For example, "Twinkle Twinkle, Little Star" begins with the notes Do and Sol (1 and 5 of the scale), while "Three Blind Mice" begins with Mi, Re, Do (3, 2, 1).
If you're singing a cappella, you can choose any pitch to be Do, and sing the song in relation to that note. If you're familiar with the relative sounds of the notes of the scale, then matching them up with the notes on the lines and spaces of the staff becomes very logical. It takes a little practice to do it well, but it doesn't take nearly the musical knowledge that reading music on an instrument does.