  Let Y be a real Enriques surface, _2Br(Y) the subgroup of elements of order 2
of Br(Y), and s, s_{or}, and s_{nor} the number of all connected, connected
orientable, and connected non-orientable components of Y(R) respectively. Using
universal covering K3-surface X of Y, we connect dim _2Br(Y) with the s, s_{or}
and s_{nor}. As a geometric corollary of our considerations, we show that s \le
6 and s_{nor} \le 4.
