There are several definitions of real reductive groups, sometimes subtly inequivalent. The following come to my mind:
A closed subgroup of GL(n,C) closed under conjugate transpose.
The set of real points G(R) of a real algebraic group such that G(C) is reductive.
Knapp's definition i.e. a Lie group G with a reductive Lie algebra g, a Cartan decomposition g = k + p, a maximal compact subgroup K such that G = K.exp(p) and such that every automorphism of g of the form Ad(g), g in G, is in Int(g^C).
Harish-Chandra definition i.e. 3. plus the condition that the connected component of the identity of a semisimple factor of G has finite center.
Maybe there are others too. What are the relations between these different definitions?