Motivated by some properties satisfied by Gorenstein projective and
Gorenstein injective modules over an Iwanaga-Gorenstein ring, we present the
concept of left and right $n$-cotorsion pairs in an abelian category
$\mathcal{C}$. Two classes $\mathcal{A}$ and $\mathcal{B}$ of objects of
$\mathcal{C}$ form a left $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$
if the orthogonality relation $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$
is satisfied for indexes $1 \leq i \leq n$, and if every object of
$\mathcal{C}$ has a resolution by objects in $\mathcal{A}$ whose syzygies have
$\mathcal{B}$-resolution dimension at most $n-1$. This concept and its dual
generalise the notion of complete cotorsion pairs, and has an appealing
relation with left and right approximations, especially with those having the
so called unique mapping property.
The main purpose of this paper is to describe several properties of
$n$-cotorsion pairs and to establish a relation with complete cotorsion pairs.
We also give some applications in relative homological algebra, that will cover
the study of approximations associated to Gorenstein projective, Gorenstein
injective and Gorenstein flat modules and chain complexes, as well as
$m$-cluster tilting subcategories.