Welty's "A Worn Path" is about indifferent social and political forces acting on an individual, and the path that individual takes represents her life and experience and existence.
Notice that no one is really out to get her or put her down and make the journey difficult for her, but nobody really tries to help her either. This is emblematic of her existence.
The path is difficult for her, even though no one, and especially nature, tries to make it difficult. She must go under fences, up hills, and over water. She is "in over her head," as the cliche goes.
And why must she take this path? Because a little boy made an innocent mistake--he swallowed lye. If you don't know what lye is, thiink crystal Drano--that's what it's made of. The lye burnt his throat and he will die without the medication the old woman travels to town to get.
The path is the central symbol of the story--thus the name. It's what the woman must suffer to keep the boy alive.