Horizontal and vertical progression seems to mean a lot of different things to a lot of different people. I caught wind of Word of Shadow's definition and was going to write a comment. But looking around the Internet, it felt as if I needed more explanation for my thoughts. Personally I was focusing my definition too narrowly, but I have my reasons, as I will explain. I wasn't thinking about it from a high level point of view, but let's start there.
In the broadest sense, if combat were the only viable path to end-game, then the system would be vertical. Being forced into leveling and combat feels confining for some people, and they'd prefer a "horizontal system". I really don't know what that means. I've seen definitions ranging from a player-skill only system to a system in which there are multiple progression paths to end-game--combat isn't the only means to the ends.
To me, horizontal progression is more of a hybridization technique. If you have a player-skill system, players are still going to min-max implicitly and pick a role. If you have multiple paths to the end game, you are still going to pick one and vertically climb the ladder. A horizontal system means that as you progress, whether through leveling and talents or through some other means, you gain skill equally across all roles.
I was originally thinking of horizontal progression in a very specific realm of play--namely combat. The reason for this is that if you have more horizontally progressed characters, you have more true hybrids. Having the same character being able to fill multiple roles more easily facilitates grouping. If every class is not only a damage dealer, but also a healer or a tank, and they progress horizontally preserving their hybridization, then you don't have to wait around for hours looking for a tank and a healer. You are at worst looking for a single role, which half of the players online can fulfill.
Perhaps an example is necessary. Let's take a look at WoW's Druid. This is supposedly a hybrid class. If you ask your feral Druid friend to shift out of Bear form and heal for this dungeon run, he will laugh at you. "I thought you were a hybrid," you retort. "I'd have to respec," responds your friend. Where is the hybridization? It was lost in the vertical progression scheme of the talent trees. WoW Druids aren't hybrids; very few classes in WoW are hybrids.
Vertical progression is much easier to design than horizontal progression. Make a tanking tree, a healing tree, and a damage tree; done. We're going to switch gears from Druid to Death Knight because Druids are too far gone to be salvaged. Death Knights are "tank and damage dealing hybrids" yet they need to spec for 1 and only 1 role. Their talent trees are labeled Blood, Frost, and Unholy. From a tanking perspective, Blood gives more health and healing powers, Frost gives more armor, and Unholy helps to reduce burst damage.
There are talents in these 3 trees which are specifically for either tanking or for doing damage, but very few for both. If we treat Blood, Frost, and Unholy as different flavors or play styles of the same class, we have an easier time thinking about horizontal progression. Blood will have a vampiric and sacrificial feel about it. Frost will concern itself with slowing and nuke capabilities. And Unholy will deal with damage over time and other debuffs. Thus, as a player spends talent points in the tree (and play style) he chooses, he will gain power equally in both tanking and damage capabilities. He will be unable to min-max into a role.
Now all Death Knight characters, regardless of what specialization the player chooses, will be a tank and damage dealing hybrid. If your group needs a tank, invite a DK and ask him to tank. The player must rely solely on his player-ability to tank, not his gear nor spec. He has progressed his character explicitly down a horizontal path of his choosing and is not gimped in any way because of it.