In this thesis we consider the dynamics of the two-parameter Lozi family of planar homeomorphisms, more precisely, the relationship between the stable and unstable manifold of the hyperbolic fixed point \(X\) of that family in the first quadrant together with their intersections, homoclinic points. We describe the zigzag structure which the stable manifold of \(X\) forms in the third quadrant, prove that all homoclinic points in the border case are tangential and construct polygons bounded by the stable and unstable manifold of \(X\) which allows us to conclude that all border homoclinic tangencies consist of iterates of two distinct points \(T_0\) and \(V_0\) which are the points at which the stable and the unstable manifold, starting from \(X\), intersect the horizontal and vertical axis for the first time respectively. In addition, by posing analytic conditions on the stable and unstable manifold of \(X\), we compute the equations of the first few curves representing the border of the set of existence of homoclinic points for that fixed point. The determined border is utilized for the introduction of a region in the parameter space for which there are no homoclinic points for \(X\) and the period-two cycle is attracting. In this region we further investigate the unstable manifold of \(X\), construct polygons which are in part bounded by it and invariant under the square of the Lozi map and in addition, prove that every part of the unstable manifold which is a finite polygonal line has an open neighborhood disjoint from its complement. We ultimately prove that the topological entropy of the Lozi map is zero for all parameter pairs in the mentioned region if the unstable manifold of \(X\) intersects the coordinate axes at \(T_0\) and its inverse image only. Along with this result, we also show that the topological entropy is zero on the complement of the accumulation set of the unstable manifold of \(X\). These results expand the already known ones about the zero entropy locus by a large set of parameters. In addition, these results are used to observe the basin of attraction for the Lozi map. We turn our attention to the stable manifold of the fixed point \(Y\) in the third quadrant: we prove that it intersects the horizontal axis at a point right to \(T_0\) which implies that it tends to infinity and accumulates on the stable manifold of \(X\) in the first quadrant. As a consequence, the stable manifold of \(Y\) separates the plane into two connected components and we prove that the basin of attraction is contained in one of them.