Closed subsets and closed subschemes Consider a scheme $(X,\mathcal O_X)$; a closed subscheme of $(X,\mathcal O_X)$ is a scheme $(Z,\mathcal O_Z)$ such that:


*

*$Z$ is a closed subset of $X$

*There is a morphism of schemes $(j,j^{\sharp}):(Z,\mathcal O_Z)\longrightarrow (X,\mathcal O_X) $ where $j$ is the topological immersion of $Z$ (with the induced topology) in $X$


Now given a closed subset $Z\subseteq X$, one may define many structures of closed subscheme on $Z$, but the question is the following:

Is there at least one structure of closed subscheme on $Z$?

Moreover, geometrically I can't imagine a closed subvariety with two structures of closed subschemes.
Thanks in advance 
 A: For any scheme and any closed subset, one can always consider the reduced induced subscheme structure. More generally, this works for any locally closed subscheme. For the first statement, one can consult Example 3.2.6 in Hartshorne. The second statement is in the book by Goertz-Wedhorn.
Consider the affine case $X = \text{Spec}(A)$. Then the closed subschemes of $X$ correspond to ideals $I \subset A$. If $\text{rad}(I) = \text{rad}(J)$, then $I$ and $J$ then $\text{Spec}(A/I)$ and $\text{Spec}(A/J)$ are (or more accurately, can be identified with) the same subsets of $\text{Spec}(A)$ (think about what it means to be a prime in $A/I$ and $A/J$), but if $I \neq J$ then they have different scheme structures. In this example, $\text{Spec}(A/\text{rad}(I))$ is the reduced induced closed subscheme structure on the set $\text{Spec}(A/I)$.
To make this really explicit, think about $X = \text{Spec}(\mathbb{C}[x])$ and consider the ideals $(x)$ and $(x)^2 = (x^2)$ in $\mathbb{C}[x]$. Both of them correspond to the point "$x = 0$" but the scheme structure (the "geometry") is different. You can see this by thinking about functions on $Z_1 = \text{Spec}(\mathbb{C}[x]/(x))$ vs. functions on $Z_2 = \text{Spec}(\mathbb{C}[x]/(x^2))$. For a function $f \in \mathbb{C}[x]$ (i.e. a function on $X$), when we look at it as a function on $Z_1$ we only see the constant term. But as a function on $Z_2$ we also get to see the linear term $a_0 +a_1x$.
For much more on this, I highly recommend the book "Geometry of Schemes" by Eisenbud and Harris.
Finally, you mention that you can't imagine a closed subvariety with more than one scheme structure. This is because a variety is (by definition) reduced, and thus there is only one scheme structure.
