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Let $X$ be a scheme and $x\in X$ a point. The stalk of $X$ at $x$ in the Zariski topology is the local ring $\mathcal{O}_{X,x}$. The stalk of $X$ at $x$ in the étale topology is the strict henselization $\mathcal{O}_{X,x}^{sh}$ of the local ring $\mathcal{O}_{X,x}$ (Wikipedia).

What is an algebraic description of the stalk of $X$ at $x$ in the fppf topology?

To define algebraic description I can only say what it's not. It shouldn't mean something like: ''the stalk of $X$ at $x$ in the fppf topology is the direct limit of the sections of the structure sheaf over all the fppf neighborhoods of $x$''. Probably the question could be rephrased as: ''How can I compute the stalk of $X$ at $x$ in the fppf topology?''

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This appears to be an open problem: see question 42258 at MO. However, Wraith [1979, Generic Galois theory of local rings] speculated that the fppf local rings are ‘algebraically closed local rings in an appropriate sense’:

Since a ring of polynomials in many variables is finitely presented and faithfully flat over its subring of symmetric polynomials, one may deduce that the inverse image of the generic commutative ring in the fppf topos has the property that monic polynomials over it split into linear factors.

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