In at least the univariate case, the guess of "Newton-Raphson" is not too far off. Daniel mentions the three possible methods supported by
NRoots in his comment.
"CompanionMatrix" is likely done by forming the Frobenius companion matrix from the polynomial's coefficients, and then performing the usual QR algorithm to extract the eigenvalues, so that is the odd one out. (There are more efficient structure-exploiting methods than vanilla QR (e.g. this), and other possible companion matrices (e.g. this), but it doesn't seem to me that these specializations are exploited.)
The other two are basically specially-adapted Newton-Raphson methods, with somewhat better convergence properties. Jenkins-Traub, which was the default algorithm in older versions of Mathematica (I don't know now), can be interpreted as either a Newton-Raphson method applied to a specially constructed rational function, or a modified Rayleigh quotient iteration of the Frobenius companion matrix (making it a direct descendant of the classical Bernoulli method). A full description is rather involved, so I'll just link to these two papers.
Aberth's method, on the other hand, is one of a family of "simultaneous iteration" methods that are based on treating the classical Vieta formulae as a set of simultaneous nonlinear equations relating the roots and coefficients, and applying Newton-Raphson to that nonlinear system. The classical reference is this paper. (I demonstrated the derivation of the related but simpler (Weierstrass-)Durand-Kerner iteration in this math.SE answer.)
A rather comprehensive survey of all these methods is McNamee's two books.