We investigate the structure of the free p -convex Banach lattice {\mathrm{FBL}}^{(p)}[E] over a Banach space E . After recalling why such a free lattice exists, and giving a convenient functional representation of it, we focus our study on how properties of an operator T:E\rightarrow F between Banach spaces transfer to the associated lattice homomorphism \bar{T}:{\mathrm{FBL}}^{(p)}[E]\rightarrow{\mathrm{FBL}}^{(p)}[F] . Particular consideration is devoted to the case when the operator T is an isomorphic embedding, which leads us to examine extension properties of operators into \ell_{p} , and several classical Banach space properties such as being a G.T. space. A detailed investigation of basic sequences and sublattices of free Banach lattices is provided. Among other things, this allows us to settle an a priori unrelated question, providing the first instance of a subspace of a Banach lattice without bibasic sequences. In addition, we begin to build a dictionary between Banach space properties of E and Banach lattice properties of {\mathrm{FBL}}^{(p)}[E] . In particular, we characterize the existence of lattice copies of \ell_{1} in {\mathrm{FBL}}^{(p)}[E] and show that \mathrm{FBL}[E] has an upper p -estimate if and only if \textup{id}_{E^*} is (q,1) -summing ( {1}/{p}+{1}/{q}=1 ). We also highlight the significant differences between {\mathrm{FBL}}^{(p)} -spaces depending on whether p is finite or infinite. For example, we show that \mathrm{FBL}^{(\infty)}[E] is lattice isometric to \mathrm{FBL}^{(\infty)}[F] whenever E and F have monotone finite-dimensional decompositions, while, on the other hand, when p<\infty and E^{*} is smooth, {\mathrm{FBL}}^{(p)}[E] determines E isometrically.