A Hausdorff topological space $(X,\mathcal T)$ is called H-closed or absolutely closed if it is closed in any Hausdorff space which contains $X$ as a subspace.
Remark: H-closedness is not hereditary with respect to closed subsets.
Theorem: A regularly closed subset of an H-closed space is H-closed.
What is the necessary and sufficient condition for a subset of an H-closed space to be H-closed?
I do not know of a non-tautologous necessary and sufficient condition for general Hausdorff spaces, but for Urysohn spaces the problem is easier. The 'nice' Hausdorff spaces are all Urysohn, but the answers to this question give examples of spaces which are Hausdorff but not Urysohn.
Here is a condition given in Proposition 3.2 (ii) in the paper Closed Subspaces of H-Closed Spaces by Johannes Vermeer.
Let $X$ be an H-Closed Urysohn space. Then $A$ is an H-closed subspace of $X$ iff $A$ is a compact subspace of $X_s$ and the restriction of the map $\text{id}:X_s\rightarrow X$ to $A$ is $\theta$-continuous.
Definitions
A Urysohn space is a space in which any two distinct points have distinct closed neighbourhoods.
$X_s$ is the semiregularisation of $X$ i.e. the smallest space which contains $X$ whose regular open sets form a base.
A function $f:X\rightarrow Y$ is theta-continuous if for every $x\in X$ and neighbourhood $U$ of $f(x)$, there exists a neighbourhood $V$ of $x$ such that $f(\text{cl}(V))\subset \text{cl}(U)$.
