Vertical organisation structures break the organisation into several semi-autonomous units, each responsible for a product line, customer-segment or region, and each with its own functional teams, such as manufacturing sales, finance, HR, etc.
Horizontal organisation structures group teams by functional disciplines, with a single sales, a single marketing, a single finance and a single HR function, etc. working across the entire organisation.
Hybrid organisations might have core functions such as manufacturing, sales and marketing arranged vertically, and support functions, such as finance and HR arranged horizontally.
Finally, matrix organisations include both vertical and horizontal structures across all functions, with each team member having two bosses: one in the vertical structure and one in the horizontal structure. As you can imagine, having two bosses can complicate things considerably, so matrix structures often given one boss priority over another (with the subordinate relationship often described as a 'dotted' reporting line, giving the matrix a horizontal or vertical flavor.
Of course, few organisations structures are pure, and many contain elements from more than one of the above-mentioned types.
My own experience is that the most successful, and the most politically astute people have a very keen sense of organizational structure, but are also able to operate around and outside of it to achieve their objectives. That is, they know how to break the rules. Not in the sense of committing fraud, or being willfully insubordinate, but in the sense of understanding how power flows through the hierarchy, and how to wield influence.
The formal organisation structure exists to manage the organisation in its stable designed state. To effect substantial and/or strategic change you need to understand how the really important decisions get made. And regardless of formal hierarchy, there are always some people who are able to wield more influence than others. These are the people who have learned to lead through influence, in addition to through formal authority. These are the people everyone looks to in times of uncertainty: the go-to people, who always seem to be one step ahead of the pack.
To effect strategic change you need to understand the formal organisation structure, but also the informal patterns of influence and power; and you need to build strong relationships with those key individuals.