The short answer, I think, is that a song seems great where there is a meeting between excellent subject matter and listener receptivity.  Sounds odd, I know, but here's why I think this is an accurate way to think of the matter.

What I regard as great songs have one thing in common. They create a powerful emotional response. All of the elements listed by the last poster (see below) may have a part in that, but even when a song is very -- even starkly -- simple it can strike me as great if it evokes an answering chord in my heart.  Of course, our own personal inner workings are part of the equation. There are times, seasons of life, when you are more attuned to some types of music and lyrical messages.

Nostalgia, for one, can be a powerful emotional experience and songs that trigger that, can strike one as utterly outstanding. In this vein I would give as examples:

Time in a Bottle
Yesterday
Those were the Days, My Friend

Probably if I had heard these songs when I was not experiencing longing for the past, they would have struck me as merely sentimental. 

Love songs, especially when you are in love or longing for love, can be incredibly powerful, out of all proportion to their musical virtues. When they capture what you are feeling or trying to express they are pure gold. Examples are numerous, but a few that moved me are as follows:

The Love Theme from Romeo and Juliet
I'll have to say I Love You in a Song
Something
Cats in the Cradle (parental love)

Again, if I were disillusioned when I heard these, they might have struck me as silly, but I heard them when I was ready to be impacted by their sentiment.

Ballads that tell of great adventure or tragedy can be very moving.  Three that come to mind are as follows:

The Hangman and the Papist
The Wreck of the Edmund Fitzgerald
Ode to Bobbie Joe

Ballads tend to be more timeless than other songs, I think. Humans love to tell and hear stories. A good ballad I would appreciate at any time in life.

Some of the greatest songs I've ever heard are the found in old hymnals. For example:

A Mighty Fortress is our God
Let Heaven and Earth Combine
Joy to the World
Silent Night

There are many more. These songs move my heart in extraordinary ways. Clearly, though, they represent an example of great song writing that is not likely to impress listeners who are not receptive to thoughts about God.

For all of these reasons, it seems to me that great songs involve a meeting of exceptional content with receptivity in the hearer.

Great question

Music = The feeling of the song. Does it engage my soul? When the music is great my soul is engaged while hearing the melodies come together. Creativity, build up, variations

Vocals = Delivery, voice, usage of voice,  feeling, flow, creativity, playfulness (very artist dependant), variations, flexibility

Atmosphere = Intro, outro, vibe, ambience, pacing, changes (tempo, intensity, change of emotions)

Musical quality = How it all comes together into one symphony. It's the same principle regardless of genre

Lyrics = Creativity, story, playfulness, wordplay, unpredictability/twists, concepts /ideas

When this + more or other things come together it makes a song great. When an artist makes songs like this repeatedly with a plethora of songs, + NOT just using a predictable formula, the artist is great in my book. But there are always a few exceptions that are great in ways that break my own system of grading the music I love to hear and those are the ones I listen to the most because their music engage my soul the most

Take care
Love