When do sheaves deform over a family? 
Let $\mathcal{X} \to B$ be a flat family with some fibre $X_b \to b$.  Suppose I have a coherent sheaf $F_b$ on $X_b$.  When does it spread out to a sheaf $\mathcal{F}$ on $\mathcal{X}$ flat over $B$?  

What about a subscheme $z \subset X_b$?  Arbitrary diagrams of sheaves?  
(I am only concerned with the case where everything is defined over $\mathbf{C}$, and moreover in the local case where $B$ is a disc and I am perfectly happy to take $\mathcal{X}$ to be affine.  However $\mathcal{X}$ should not be assumed smooth, nor $X_b$ to even be reduced.) 
 A: Suppose that $B_n$ it the $n^{\rm th}$ infinitesimal neighborhood of $b$ in $B$; that is, if $\frak m$ is the maximal ideal of $b$ in $B$, we set $B_n := \mathop{\rm Spec} \mathcal O_B/{\frak m}^{n+1}$. If $\mathcal F_n$ is as extension of $\mathcal F_b$ to $B_n$, there is a canonically defined element of $({\frak m}^{n+2}/{\frak m}^{n+1})\otimes_{\mathbb C}\mathop{\rm Ext}_{\mathcal O_{X_b}}(\mathcal F, \mathcal F)$, called the obstruction; if this is zero, then the sheaf $\mathcal F_n$ extends to $B_{n+1}$. This depends on $\mathcal F_n$, not only on $\mathcal F_b$.
If this obstructions are always 0 (for example, if $\mathop{\rm Ext}_{\mathcal O_{X_b}}(\mathcal F, \mathcal F) = 0$), then $\mathcal F_b$ will extend to some étale neighborhood of $b$ in $B$. I don't think you can say much more in this generality.
There is a whole subject devoted to the study of this kind of problems (not only for sheaves, but for much more general objects), called deformation theory.
A: If $\mathcal{F}_b$ is an invertible sheaf and $B=\textrm{Spec} \mathbb{C}[\epsilon]/(\epsilon^2)$ (first-order deformations) then the obstruction theory for deforming $\mathcal{F}_b$ described in Angelo's answer becomes very explicit. Indeed, there is the following result, whose proof can be found in Sernesi's book "Deformation of algebraic schemes", p. 147.
Set $\mathcal{L}:=\mathcal{F}_b$, $X:=X_b$.
THEOREM Given a first-order deformation $\xi$ of $X$, there is a first-order deformation of $\mathcal{L}$ along $\xi$ if and only if 
$\kappa(\xi) \cdot c(\mathcal{L})=0$.
Here $\kappa$ is the Kodaira-spencer map, $c$ is the first Chern class and "$\cdot$" denotes the composition
$H^1(X, T_X) \times H^1(X, \Omega^1_X) \to H^2(X, T_X \otimes \Omega^1_X) \to H^2(X, \mathcal{O}_X)$,
where the first arrow is induced by the cup-product and the second one by the duality pairing $T_X \otimes \Omega^1_X \to \mathcal{O}_X$.
Using this, one can prove for instance that if $X$ is an Abelian variety of dimension $g$ and $\mathcal{L}$ is an ample line bundle, then $\mathcal{L}$ extends along a subspace of $H^1(X, T_X)$ of dimension $g(g+1)/2$. For $g \geq 2$, $\mathcal{L}$ does not extend to the whole of $H^1(X, T_X)$ (not even to the first-order!), since the general deformation of $X$ is not projective. 
