Your description is off a bit.
Skills ARE gems in PoE. Any class can use any skill gem, but some have stat requirements that might make them impractical. There are also support gems which modify skills, say splitting your flaming arrow into multiple shots or giving your minions more life.
To use a support gem with one or more skills, you have to have an item with sockets that are linked together, and the sockets have to match. The color of the gem (red-strength, green-dex, or blue- Int). As an example, I have a witch character I'm playing right now. I have both Summon Zombies and Summon Skeletons in an item with three linked blue sockets, along with a blue support gem (minion life) which increases the health of minions for both skills.
There is also a MASSIVE passive skill tree. All characters share the same tree, but class determines your starting location. Many of the nodes on the tree are simple things like stat boosts, but there are game changers around the fringes, things like making your attacks always hit but never do criticals, or giving you increased damage at point blank range but reducing your overall range, making minions explode at low health, making your skills use health instead of mana, etc.
The coolest thing about the game, though, is the item fabrication. There are a bunch of different crafting items called orbs in the game. These modify items in various ways, like randomizing the number or color of sockets an item has, making a white item magical or even rare, adding additional properties to items, randomizing the properties of items etc.
One of the neat side effects of this is that white base items (that you never want to see in diablo past level two) can be extremely valuable. Find a good quality white item with the sockets you need? Excellent! Drop an orb on it and make it rare? Don't like the results? Use a different orb to re-roll the item.
I don't have any high-level characters, but even at low levels, I've messed with this a bit, and it's rather cool.