It was shown at beginning and end of the Forrest Gump (1994).
What's the significance of the white feather?
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The feather reflects many of the film's messages: life is full of uncontrollable events, innocence is lighter than air, some things are the result of luck, and the fact we never know what is next on the horizon for us. "Mrs Gump: Life's a box of chocolates, Forrest. You never know what you're gonna get."
From Screen Rant, it's Forrest's approach to life:
It just floats through the world in whichever direction the wind takes it. A lot of what happens is up to chance — for good and for bad.
It's the same according to Bennett Caplin:
The feather that floats through the air at both the beginning and end of the movie is a perfect metaphor: Forrest never knows where his life will take him next, but he goes with the flow and he finds happiness at all stops along the way because that is what his mother taught him to do.