Those are the definitions given to us during our lecture:
Zariski Topology
We call a Zariski Topology in $\mathbb{K}^n$ a family of complements of $V\left(I\right)=\left\{a\in \mathbb{K}^n\::\:\forall _{f\in I}\:\:f\left(a\right)=0\right\}$, where $I\subset \mathbb{K}\left[X_1,\:...,\:X_n\right]$ is an arbitrary ideal. In other words, the algebraic sets $V\left(I\right)$ are a family of closed sets
Prime Spectrum
Let R be an arbitrary ring. For any ideal $I\subset R$ we define the set $V\left(I\right)=\left\{p\in Spec\:R\::\:I\:\subset p\right\}$ as the prime spectrum
What I'd like to know is what the motivation behind those concepts is, why were they created in the first place? What's their use, and how can I interpret them (e.g. geometrically or visually)?