If $\mathcal{E}$ is some quasi-coherent $\mathcal{O}_S$-module, the functor $(\mathrm{Sch}/S)^{\text{op}} \to \mathrm{Set}$ which sends $f : T \to S$ to the set of invertible quotients of $f^* \mathcal{E}$, is representable by a (separated) scheme $\mathbb{P}(\mathcal{E})$ over $S$. If $\mathcal{E}$ is locally free, this is called the projective space bundle associated to $\mathcal{E}$. The intuition is: Replace fiberwise $\mathbb{A}^n$ by $\mathbb{P}^n$. You can find this functorial approach, more generally for Grassmannians, in EGA I (1970), §9.

As for the notation $H^0(\mathcal{F}) \otimes \mathcal{O}_S$: This is an *external* tensor product. If $M$ is some $R$-module and $\mathcal{E}$ is some quasi-coherent $\mathcal{O}_S$-module, where $S$ is an $R$-scheme, then $M \otimes_R \mathcal{E}$ is defined either locally in the obvious way, or by the obvious universal property, or by the rule $R^n \otimes_R \mathcal{E} = \mathcal{E}^n$ and cocontinuous extension.