I'm an instrumentalist who only recently (in my mid-50's!) began to work with an amazing voice teacher, and am beginning to get a handle on my singing "instrument". So I'm not a "natural", and have been paying very close attention to the physical sensations of singing, and your question brought up a number of things.
"Effortless"? I've been taught that I should be tired after singing well for a while - it's very physical, after all - but I know what you mean - when things are flowing you're not fighting yourself, and there is no tension (muscles working against each other unnecessarily). Maybe it helps to talk about the difference between "tension" and "support" (as in "supporting the tone" in another answer here) - But since support in singing inevitably involves fantastically complex interactions of little muscles all through your body, often working "against" each other, it can be hard to tell the difference!
But it sounds like your are looking for concepts, so much as very specific, reliable techniques. One way I've learned to distinguish between "tension" and "support" is that I can voluntarily give up support, but tension takes some touching, etc. to shed. For a lot of singers, tension tends to collect in the throat and jaw. A nice trick is to run your hands down either side of your face while singing an "ah" - is your jaw hanging loose? Can you let your tongue hang out while doing the same thing? I was taught to sing "the...uhhhh" up the scale, feeling my tongue relaxed.
I was singing in a strenuous concert just two nights ago, and found that purposely clenching and releasing my butt was a great way to relax my legs - and teachers will tell you never to lock your knees, so there you go, there's a lot you can do from the waist down as well. But you probably knew about that...
One very specific thing I'll leave you with, from my life as a professional instrumentalist, is to find an Alexander Technique teacher you like (http://en.wikipedia.org/wiki/Alexander_technique). They can help you acquire an exquisite sense of your body and the way it carries itself through the slings and arrows of life - and there's some solid research behind why it works! Good luck to you, and in spite of all of this, never forget that, when you're performing, "you are the music, while the music lasts...".