In 1970, at a concert in London, Joni Mitchell claimed that she wrote this song in response to Neil Young's song "Sugar Mountain." Young wrote the song about lamenting losing his youth: specifically in response to turning 21 and being too old to go to his favorite club. Mitchell said she wrote "The Circle Game" for Young: the songs have similar themes. (Interesting to note that the song is about youth and time passing and that she claimed to have written it for "Young.") 

"The Circle Game" is about youth, nostalgia, regret, hope, and the passage of time. The tone and mood fluctuate between nostalgia, regret, and hope. The first stanza/verse starts at the beginning of life. The child is initially full of wonder. It is a rural, natural scene characteristic of the Romantic poets (i. e. Wordsworth and Keats) where the child is emotionally affected by his natural surroundings. The chorus establishes the main theme of the song/poem: the passage of time. Mitchell uses the metaphor of a carousel to describe the passage of time. She notes how the seasons "go 'round and 'round" like a carousel. The ponies on the carousel go up and down like a vertical, ticking clock. 

And the seasons they go 'round and 'round

And the painted ponies go up and down

We're captive on the carousel of time

This imagery illustrates time as something cyclical. The up and down motion of the ponies is like a clock, but the main image is that of the circle. The chorus continues, adding that we can not go back (in time) but we can look back at the past: 

We can't return, we can only look behind

From where we came 

Mitchell uses the circle, the carousel to illustrate this idea with a visual image. We can not reverse the carousel, but since it travels in a circle we can revisit where we've been at a later time; this is akin to reflecting on the past, be it with regret or nostalgia. 

Using the image of the carousel signifies the time of youth and innocence (children tend to ride carousels); hence, the title being "The Circle Game" rather than something like "The Circle of Time." So, there is a playfulness and that seems to suggest that the speaker is nostalgic for that time of youth and/or laments the loss of youth. We can not go back to that time; the best we can do is reflect upon it. 

In the second verse, the child is ten years old/older. Mitchell uses images of nature to indicate the passing of time. The child has skated over "ten clear frozen streams" indicates the child has experienced ten winters. The child is still young enough to entertain grand dreams of the future. 

In the next stanza, the child is now an adolescent. Having experienced 16 springs and summers, he is 16 and old enough to drive. "Cartwheels turn to car wheels" - Mitchell is consistent with the circle imagery, noting that the child doing cartwheels has turned into a young adult preferring the car. As this young man matures, he starts to become aware of the passage of time. He is warned (return to the carousel image) to drag his feet, as if he could slow down time/the motion of the carousel. 

In the next stanza, the boy/man is now twenty. There is more circular imagery as "the years spin by." The boy's dreams may not be as grand as in his youth, but that nostalgia/regret ends on a hopeful note: new dreams might be better "before the last revolving year is through." Development of identity is part reflecting on the past and part forging ahead into the future: like going in a circle.