  Let $\mathcal{A}$ be an abelian category. For a pair
$(\mathcal{X},\mathcal{Y}$ of classes of objects in $\mathcal{A},$ we define
the weak and the $(\mathcal{X},\mathcal{Y})$-Gorenstein relative projective
objects in $\mathcal{A}$. We point out that such objects generalize the usual
Gorenstein projective objects and others generalizations appearing in the
literature as Ding-projective, Ding-injective, $\mathcal{X}$-Gorenstein
projective, Gorenstein AC-projective and $G_C$-projective modules and
Cohen-Macaulay objects in abelian categories. We show that the principal
results on Gorenstein projective modules remains true for the weak and the
$(\mathcal{X},\mathcal{Y}$-Gorenstein relative objects. Furthermore, by using
Auslander-Buchweitz approximation theory, a relative version of Gorenstein
homological dimension is developed. Finally, we introduce the notion of
$\mathcal{W}$-cotilting pair in the abelian category $\mathcal{A}$, which is
very strong connected with the cotorsion pairs related with relative Gorenstein
objects in $\mathcal{A}$. It is worth mentioning that the
$\mathcal{W}$-cotilting pairs generalize the notion of cotilting objects in the
sense of L. Angeleri H\"ugel and F. Coelho.
