This is a very broad category, I know. Consider it on the meta-level, though. This isn't about individual skills or abilities in a game, but about how groups of them are organized or interact. First though, let's define for clarity.
A skill is an action that an avatar is capable of. This is distinct from the player's own skill or ability at playing the game. In many games these are treated as distinct concepts from any inherent skills the avatar has, such as running or jumping. I'd venture this is simply because of the need to define more complex concepts that some skills require, and because it helps train the player in the skill if it's conceptually distinct.
The term ability is typically used interchangably with skill. I'd like to define it as "modifying an avatar's attributes". But since most gamers use them interchangably, I'll do the same for now and explicitly say when I'm using it in that context.
In many games, skills are typically a way of layering the basic gameplay with more complexity. This is how something like Dead Island treats it. On the other hand, consider skill trees, as presented in Diablo or World of Warcraft. The player is awarded a level, along with a skill point, which can be either hoarded or spent upon skills. Typically these skills are split along multiple branches. It's also typically beneficial to invest fully down a single branch given that they last skills in that branch are the most powerful, and reaching them quickly confers a strong advantage.
In Blizzard's World of Warcraft, it should be noted, expert gamers have gone to great lengths, reverse engineering (through calculus and simulation) the skill trees to determine the maximum value (DPS in this case) possible with given skill trees. This effectively means that although the trees may give the appearance of choice, there is truly a most effective choice. This means any expert player in the game will select these builds, limiting any variety the trees originally were supposed to engender.
- Fighters: usually these are inherent in the different characters, and require player skill to execute avatar skill.
- RPGs: Skills have been experimented on massively in this genre, and vary radically.