Geometric properties can often be described as properties of $B(X)$ (or subspaces of it) or as properties on $X$. (Of course we only call them 'geometric' if we can describe them in $X$). Sometimes geometric properties were first observed as properties of operators. Let me give an example:
In 1964, if I'm not mistaken, Daugavet observed that on $C([0,1])$, every compact $T\in K(C[0,1])$ (denoting by $K(X)$ the space of compact operators on $X$) has maximal distance to the identity, that is we have \[ \|\mathrm{Id} + T\| = 1 + \| T\|, \quad T \in K(C[0,1]) \] This property seems first to be a property of operators, but intrestingly, this property, which was afterwards called the Daugavet property, can be described geometrically. A space $X$ is said to be (a) Daugavet (space), if it shares this property. Now for the geometric description: The part of the unit ball $B_X$ which lies on one side of a hyperplane is called a slice, that is a slice a set (for $x^* \in S_{X^*}$ and $\epsilon > 0$) \[ S(x^*,\epsilon) = \{x \in B_X \mid x^*(x) \ge 1 -\epsilon \} \] A space is now exactly then Daugavet iff these slices have 'big' diameter, that is given a slice $S(x^*, \epsilon)$ an $y \in S_X$, and a $\delta > 0$, there is an $x \in S(x^*, \epsilon)$ with $\|x-y\| \ge 2 - \delta$.