A Euclidean supermanifold is a supermanifold that can be thought of as being equipped with a flat Riemannian metric.
Alternatively, it is a supermanifold for which the transition functions of an atlas are restricted to be elements of the super Euclidean group.
A Euclidean supermanifold of dimension $(p|q)$is a supermanifold that is quipped with an $(X,G)$-structure , where $X = \mathbb{R}^{p|q}$ and where $G$ is the super Euclidean group on $\mathbb{R}^{p|q}$.
Here an $(X,G)$-structure is defined as follows, essentially being a version of the discussion of pseudogroups at manifold.
Definition (Stolz, Teichner) A $(X,G)$-structure on a $(d|\delta)$-dimensional supermanifold $Y$ consists of
a maximal atlas consisting of charts
(where on the left $Y_\red\supset_{open} (U_i)_{red}$) with $O_Y|_{(U_i)_{red} = O_{U_i}}$
such that the transition function
is the restriction of a map
definition A family of $(X,G)$-(complex-, super-)manifolds is a map
together with a maximal atlas of charts
such that the transition maps
are the restriction of a map of the following form
for some
example a family $Y \to S$ of $(\mathbb{R}^d, Eucl(\mathbb{R}^d))$-manifolds, for $S$ an ordinary manifold is a submersion with flat Riemannian metric on the fibers.
Specifically in 2-dimensions, an ordinary Spin-Eulidean manifold is one with $(\mathbb{R}^2, \mathbb{R}^2 \rtimes Spin(2))$-structure.
We want to regard this as a Euclidean supermanifold with $(\mathbb{R}^{2|1}_{cs}, \mathbb{R}^{2|1}_{cs} \rtimes Spin(2))$-structure.
In general for two structures $(X,G)$ and $(X',G')$ we can transfer structures when we have a group homomorphisms
$G \to G'$ and with respect to that a
$G$-equivariant map $X' \to X$.
Then send every $(X,G)$-chart to the corresponding $(X',G')$-chart which as a subset of $X'$ is the inverse image of $X' \to X$.
This yields a functor