How to determine ability + characteristic total?

I am confused...

What characteristic is used with ability?

Let says that character has Swim ability. What kind of characteristic I will use with it?

Or Legerdemain...what characteristic is used with that?

It depends on what you're doing. Take the Swim check, for example.

Are you trying to swim three miles from your sunk ship to shore? Stamina + Swim
Are you trying to swim through a small tunnel from the cave you're in to the open sea? Dexterity + Swim
Are you trying to Swim while dragging an unconscious comrade? Strength + Swim

It all depends on what the roll is trying to accomplish.

At least that's how I understand things. I'll wait to see what wiser heads than I say.

Trogdor has the right of it. Certain abilities will generally pair more strongly with other characteristics. Folk Ken is primarily perception, for example. I've done Bargain +Com for making a deal, and Bargain + Perception for assessing value.

As said above, the choice of which Characteristic adds to the Ability is arbitrary, decided by the SG (or group, depending on your playing group social dynamics) based on what the character is trying to do.

However, I suggest most often Swim and Legerdemain will use Dexterity. So if you're just trying to swim well, or pick-pocket, and am unsure what Characteristic to use, I'd suggest a Dex + Swim/Legerdemain will probably do.

Still, it depends. If you're trying to swim fast Quickness + Swim might be appropriate, if you're trying to spot another person pick-pocketing an item a Perception + Legerdemain check might work, and so on. Use the Characteristic that fits best.

I like this flexibility. Better than the One Char+Skill on other games and systems.

I agree with Trogdor in that it depends on the exact situation.
And I agree with Mario aboiut the flexibility.

Some of the earlier editions had characteristics listed for each ability. 2nd ed had two choices listed. 3rd had none. 4th had a single one.

In my troupe we often see situations where the characteristics used is a dead giveaway. Awareness rolls rarely use other than Per. Other abilties often use different characteristics depending on circumstances. The Swin example above is a good one. Sometimes the SG or the Troupe mention different possibilities and the player choses one. Other times the SG asks the player which ability+char he is using and the Trupe decides if the choice is relevant. And the scene plays out depending on this choice. Social abilities is one thing we allow a lot of leeway in.