The "normal sets" are separable metric spaces with no isolated points, as introduced by Fréchet in Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo 22, 1-74 (1906). See in particular pages 23-24, where the "classes normales" are defined as being  "parfaites, séparables et admettant une généralisation du théorème de Cauchy".
For an extensive discussion of Brouwer's paper in the historical context see D.M. Johnson's 1981 article in the Archive for History of Exact Sciences. Johnson notes that Brouwer is largely following F. Hausdorff's Grundzüge der Mengenlehre in his classification of the normal sets.
 The reference to Cauchy's theorem is the requirement that the limit of every subsequence of a sequence converging to an element $A$ is also $A$.