Let $S$ be an object in an abelian category. Then we say S is spherical if $Ext^p(S,S)$ is 0 unless $p = 3$.
I know that the cohomology of the three sphere bears some formal resemblence, but it doesn't seem strong enough to deserve this name. Why is there some relatinoship between the "cohomology" of $S$ if it was a space and $Ext^i(S,S)$?
tl;dr: Spherical objects in $D^b\operatorname{Coh}(X)$ correspond to Lagrangian spheres in $D^b\operatorname{Fuk}(M)$.
I'm not an expert in this area, but let me make a few comments anyway. First, I believe that the general definition of a spherical object doesn't require $p = 3$ in particular. We'll call an object $E$ spherical if $\operatorname{Ext}^*(E,E) \cong H^*(S^n)$ for some $n$. I'm leaving out some other conditions, but let's not worry about that.
If I recall, the notion of spherical objects first arose in Seidel-Thomas' work on the group of autoequivalences of $D^b(X)$, say for $X$ a smooth complex projective variety. Given any object $E$ in $D^b(X)$, one can define a twist functor as roughly follows. We have an evaluation map $$\operatorname{Hom}^\bullet(E,\cdot) \otimes E \xrightarrow{\operatorname{ev}} \mathrm{id},$$ and we'll define the twist functor $T_E$ to be the cone of this map. One of the main theorems in the Seidel-Thomas paper is that $T_E$ is an autoequivalence iff $E$ is spherical.
What does this have to do with spheres? For that we'll need a brief digression into symplectic geometry. Given a symplectic manifold $M$ and a Lagrangian submanifold $S$, we'll say that $S$ is a Lagrangian sphere if it is diffeomorphic to $S^n$. Given a Lagrangian sphere $S$, one can define a symplectic automorphism $\tau_S$ of $M$, the generalized Dehn twist along $S$.
Now suppose $M$ has a mirror partner $X$ (in the sense of Kontsevich). Ignoring a bunch of details, if $E$ is the object in $D^b(X)$ corresponding to the Lagrangian sphere $S \in D^b \operatorname{Fuk}(M)$, then $$\operatorname{Hom}^\bullet_{D^b(X)}(E,E) \cong \operatorname{Hom}^\bullet_{D^b\operatorname{Fuk}(M)}(S,S) \cong HF^\bullet(S,S) \otimes \mathbb{C}.$$
For Lagrangian spheres, $HF^*(S,S) \cong H^*(S;\mathbb{R})$, so $E$ must be a spherical object. In their paper, Seidel-Thomas further conjectures that the twist functor $T_E$ and the generalized Dehn twist $\tau_S$ correspond to each other under mirror symmetry. If you're interested, I'm sure the paper (which I'm paraphrasing from) would provide more insight into these spherical objects.